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L3 Lab 3

case refine_1 R : Type u ι✝ : Type u_1 inst✝ : CommSemiring R I✝ J K L : Ideal R ι : Type u_2 s : Finset ι I : ι → Ideal R x : ι → R a✝ : ∀ i ∈ ∅, x i ∈ I i ⊢ ∏ i in ∅, x i ∈ ∏ i in ∅, I i

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro

rw [Finset.prod_empty, Finset.prod_empty, one_eq_top]

theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro

Mathlib.RingTheory.Ideal.Operations.459_0.5qK551sG47yBciY

theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i

Mathlib_RingTheory_Ideal_Operations

case refine_1 R : Type u ι✝ : Type u_1 inst✝ : CommSemiring R I✝ J K L : Ideal R ι : Type u_2 s : Finset ι I : ι → Ideal R x : ι → R a✝ : ∀ i ∈ ∅, x i ∈ I i ⊢ 1 ∈ ⊤

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top]

exact Submodule.mem_top

theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top]

Mathlib.RingTheory.Ideal.Operations.459_0.5qK551sG47yBciY

theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i

Mathlib_RingTheory_Ideal_Operations

case refine_2 R : Type u ι✝ : Type u_1 inst✝ : CommSemiring R I✝ J K L : Ideal R ι : Type u_2 s : Finset ι I : ι → Ideal R x : ι → R ⊢ ∀ ⦃a : ι⦄ {s : Finset ι}, a ∉ s → ((∀ i ∈ s, x i ∈ I i) → ∏ i in s, x i ∈ ∏ i in s, I i) → (∀ i ∈ insert a s, x i ∈ I i) → ∏ i in insert a s, x i ∈ ∏ i in insert a s, I i

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top ·

intro a s ha IH h

theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top ·

Mathlib.RingTheory.Ideal.Operations.459_0.5qK551sG47yBciY

theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i

Mathlib_RingTheory_Ideal_Operations

case refine_2 R : Type u ι✝ : Type u_1 inst✝ : CommSemiring R I✝ J K L : Ideal R ι : Type u_2 s✝ : Finset ι I : ι → Ideal R x : ι → R a : ι s : Finset ι ha : a ∉ s IH : (∀ i ∈ s, x i ∈ I i) → ∏ i in s, x i ∈ ∏ i in s, I i h : ∀ i ∈ insert a s, x i ∈ I i ⊢ ∏ i in insert a s, x i ∈ ∏ i in insert a s, I i

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h

rw [Finset.prod_insert ha, Finset.prod_insert ha]

theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h

Mathlib.RingTheory.Ideal.Operations.459_0.5qK551sG47yBciY

theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i

Mathlib_RingTheory_Ideal_Operations

case refine_2 R : Type u ι✝ : Type u_1 inst✝ : CommSemiring R I✝ J K L : Ideal R ι : Type u_2 s✝ : Finset ι I : ι → Ideal R x : ι → R a : ι s : Finset ι ha : a ∉ s IH : (∀ i ∈ s, x i ∈ I i) → ∏ i in s, x i ∈ ∏ i in s, I i h : ∀ i ∈ insert a s, x i ∈ I i ⊢ x a * ∏ x_1 in s, x x_1 ∈ I a * ∏ x in s, I x

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha]

exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi)

theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha]

Mathlib.RingTheory.Ideal.Operations.459_0.5qK551sG47yBciY

theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R S T : Set R ⊢ span S * span T = span (S * T)

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by

unfold span

theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by

Mathlib.RingTheory.Ideal.Operations.522_0.5qK551sG47yBciY

theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T)

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R S T : Set R ⊢ Submodule.span R S * Submodule.span R T = Submodule.span R (S * T)

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span

rw [Submodule.span_mul_span]

theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span

Mathlib.RingTheory.Ideal.Operations.522_0.5qK551sG47yBciY

theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T)

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R r s : R ⊢ span {r} * span {s} = span {r * s}

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by

unfold span

theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by

Mathlib.RingTheory.Ideal.Operations.527_0.5qK551sG47yBciY

theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R)

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R r s : R ⊢ Submodule.span R {r} * Submodule.span R {s} = Submodule.span R {r * s}

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span

rw [Submodule.span_mul_span, Set.singleton_mul_singleton]

theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span

Mathlib.RingTheory.Ideal.Operations.527_0.5qK551sG47yBciY

theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R)

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R s : R n : ℕ ⊢ span {s} ^ n = span {s ^ n}

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by

induction' n with n ih

theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by

Mathlib.RingTheory.Ideal.Operations.533_0.5qK551sG47yBciY

theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R)

Mathlib_RingTheory_Ideal_Operations

case zero R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R s : R ⊢ span {s} ^ Nat.zero = span {s ^ Nat.zero}

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; ·

simp [Set.singleton_one]

theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; ·

Mathlib.RingTheory.Ideal.Operations.533_0.5qK551sG47yBciY

theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R)

Mathlib_RingTheory_Ideal_Operations

case succ R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R s : R n : ℕ ih : span {s} ^ n = span {s ^ n} ⊢ span {s} ^ Nat.succ n = span {s ^ Nat.succ n}

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one]

simp only [pow_succ, ih, span_singleton_mul_span_singleton]

theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one]

Mathlib.RingTheory.Ideal.Operations.533_0.5qK551sG47yBciY

theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R)

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝ : CommSemiring R I✝ J K L : Ideal R x y : R I : Ideal R ⊢ x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by

simp only [mul_comm, mem_mul_span_singleton]

theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by

Mathlib.RingTheory.Ideal.Operations.542_0.5qK551sG47yBciY

theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝ : CommSemiring R I✝ J✝ K L : Ideal R x : R I J : Ideal R ⊢ (∀ {zI : R}, zI ∈ I → zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by

simp only [mem_span_singleton_mul]

theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by

Mathlib.RingTheory.Ideal.Operations.546_0.5qK551sG47yBciY

theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝ : CommSemiring R I✝ J✝ K L : Ideal R x : R I J : Ideal R ⊢ span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by

simp only [mul_le, mem_span_singleton_mul, mem_span_singleton]

theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by

Mathlib.RingTheory.Ideal.Operations.552_0.5qK551sG47yBciY

theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝ : CommSemiring R I✝ J✝ K L : Ideal R x : R I J : Ideal R ⊢ (∀ (r : R), x ∣ r → ∀ s ∈ I, r * s ∈ J) ↔ ∀ z ∈ I, x * z ∈ J

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton]

constructor

theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton]

Mathlib.RingTheory.Ideal.Operations.552_0.5qK551sG47yBciY

theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J

Mathlib_RingTheory_Ideal_Operations

case mp R : Type u ι : Type u_1 inst✝ : CommSemiring R I✝ J✝ K L : Ideal R x : R I J : Ideal R ⊢ (∀ (r : R), x ∣ r → ∀ s ∈ I, r * s ∈ J) → ∀ z ∈ I, x * z ∈ J

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor ·

intro h zI hzI

theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor ·

Mathlib.RingTheory.Ideal.Operations.552_0.5qK551sG47yBciY

theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J

Mathlib_RingTheory_Ideal_Operations

case mp R : Type u ι : Type u_1 inst✝ : CommSemiring R I✝ J✝ K L : Ideal R x : R I J : Ideal R h : ∀ (r : R), x ∣ r → ∀ s ∈ I, r * s ∈ J zI : R hzI : zI ∈ I ⊢ x * zI ∈ J

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI

exact h x (dvd_refl x) zI hzI

theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI

Mathlib.RingTheory.Ideal.Operations.552_0.5qK551sG47yBciY

theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J

Mathlib_RingTheory_Ideal_Operations

case mpr R : Type u ι : Type u_1 inst✝ : CommSemiring R I✝ J✝ K L : Ideal R x : R I J : Ideal R ⊢ (∀ z ∈ I, x * z ∈ J) → ∀ (r : R), x ∣ r → ∀ s ∈ I, r * s ∈ J

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI ·

rintro h _ ⟨z, rfl⟩ zI hzI

theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI ·

Mathlib.RingTheory.Ideal.Operations.552_0.5qK551sG47yBciY

theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J

Mathlib_RingTheory_Ideal_Operations

case mpr.intro R : Type u ι : Type u_1 inst✝ : CommSemiring R I✝ J✝ K L : Ideal R x : R I J : Ideal R h : ∀ z ∈ I, x * z ∈ J z zI : R hzI : zI ∈ I ⊢ x * z * zI ∈ J

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI

rw [mul_comm x z, mul_assoc]

theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI

Mathlib.RingTheory.Ideal.Operations.552_0.5qK551sG47yBciY

theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J

Mathlib_RingTheory_Ideal_Operations

case mpr.intro R : Type u ι : Type u_1 inst✝ : CommSemiring R I✝ J✝ K L : Ideal R x : R I J : Ideal R h : ∀ z ∈ I, x * z ∈ J z zI : R hzI : zI ∈ I ⊢ z * (x * zI) ∈ J

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc]

exact J.mul_mem_left _ (h zI hzI)

theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc]

Mathlib.RingTheory.Ideal.Operations.552_0.5qK551sG47yBciY

theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝ : CommSemiring R I✝ J✝ K L : Ideal R x y : R I J : Ideal R ⊢ span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by

simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm]

theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by

Mathlib.RingTheory.Ideal.Operations.563_0.5qK551sG47yBciY

theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝¹ : CommSemiring R I J K L : Ideal R inst✝ : IsDomain R x : R hx : x ≠ 0 ⊢ span {x} * I ≤ span {x} * J ↔ I ≤ J

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by

simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def]

theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by

Mathlib.RingTheory.Ideal.Operations.568_0.5qK551sG47yBciY

theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝¹ : CommSemiring R I J K L : Ideal R inst✝ : IsDomain R x : R hx : x ≠ 0 ⊢ I * span {x} ≤ J * span {x} ↔ I ≤ J

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by

simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx

theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by

Mathlib.RingTheory.Ideal.Operations.574_0.5qK551sG47yBciY

theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝¹ : CommSemiring R I J K L : Ideal R inst✝ : IsDomain R x : R hx : x ≠ 0 ⊢ span {x} * I = span {x} * J ↔ I = J

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by

simp only [le_antisymm_iff, span_singleton_mul_right_mono hx]

theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by

Mathlib.RingTheory.Ideal.Operations.579_0.5qK551sG47yBciY

theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝¹ : CommSemiring R I J K L : Ideal R inst✝ : IsDomain R x : R hx : x ≠ 0 ⊢ I * span {x} = J * span {x} ↔ I = J

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by

simp only [le_antisymm_iff, span_singleton_mul_left_mono hx]

theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by

Mathlib.RingTheory.Ideal.Operations.584_0.5qK551sG47yBciY

theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝ : CommSemiring R I✝ J✝ K L : Ideal R x : R I J : Ideal R ⊢ I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by

simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff]

theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by

Mathlib.RingTheory.Ideal.Operations.599_0.5qK551sG47yBciY

theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝ : CommSemiring R I✝ J✝ K L : Ideal R x y : R I J : Ideal R ⊢ span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by

simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm]

theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by

Mathlib.RingTheory.Ideal.Operations.604_0.5qK551sG47yBciY

theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R m : Multiset R ⊢ Multiset.prod (Multiset.map (fun x => span {x}) 0) = span {Multiset.prod 0}

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by

simp

@[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by

Mathlib.RingTheory.Ideal.Operations.620_0.5qK551sG47yBciY

@[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R)

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R m✝ : Multiset R a : R m : Multiset R ih : Multiset.prod (Multiset.map (fun x => span {x}) m) = span {Multiset.prod m} ⊢ Multiset.prod (Multiset.map (fun x => span {x}) (a ::ₘ m)) = span {Multiset.prod (a ::ₘ m)}

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by

simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton]

@[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by

Mathlib.RingTheory.Ideal.Operations.620_0.5qK551sG47yBciY

@[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R)

Mathlib_RingTheory_Ideal_Operations

R : Type u ι✝ : Type u_1 inst✝ : CommSemiring R I✝ J K L : Ideal R ι : Type u_2 s : Finset ι I : ι → R hI : Set.Pairwise (↑s) (IsCoprime on I) ⊢ (Finset.inf s fun i => span {I i}) = span {∏ i in s, I i}

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by

ext x

theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by

Mathlib.RingTheory.Ideal.Operations.627_0.5qK551sG47yBciY

theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i}

Mathlib_RingTheory_Ideal_Operations

case h R : Type u ι✝ : Type u_1 inst✝ : CommSemiring R I✝ J K L : Ideal R ι : Type u_2 s : Finset ι I : ι → R hI : Set.Pairwise (↑s) (IsCoprime on I) x : R ⊢ (x ∈ Finset.inf s fun i => span {I i}) ↔ x ∈ span {∏ i in s, I i}

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x

simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton]

theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x

Mathlib.RingTheory.Ideal.Operations.627_0.5qK551sG47yBciY

theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i}

Mathlib_RingTheory_Ideal_Operations

case h R : Type u ι✝ : Type u_1 inst✝ : CommSemiring R I✝ J K L : Ideal R ι : Type u_2 s : Finset ι I : ι → R hI : Set.Pairwise (↑s) (IsCoprime on I) x : R ⊢ (∀ i ∈ s, I i ∣ x) ↔ ∏ i in s, I i ∣ x

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton]

exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩

theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton]

Mathlib.RingTheory.Ideal.Operations.627_0.5qK551sG47yBciY

theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i}

Mathlib_RingTheory_Ideal_Operations

R : Type u ι✝ : Type u_1 inst✝¹ : CommSemiring R I✝ J K L : Ideal R ι : Type u_2 inst✝ : Fintype ι I : ι → R hI : ∀ (i j : ι), i ≠ j → IsCoprime (I i) (I j) ⊢ ⨅ i, span {I i} = span {∏ i : ι, I i}

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by

rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton]

theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by

Mathlib.RingTheory.Ideal.Operations.635_0.5qK551sG47yBciY

theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i}

Mathlib_RingTheory_Ideal_Operations

case hI R : Type u ι✝ : Type u_1 inst✝¹ : CommSemiring R I✝ J K L : Ideal R ι : Type u_2 inst✝ : Fintype ι I : ι → R hI : ∀ (i j : ι), i ≠ j → IsCoprime (I i) (I j) ⊢ Set.Pairwise (↑Finset.univ) (IsCoprime on fun i => I i)

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton]

rwa [Finset.coe_univ, Set.pairwise_univ]

theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton]

Mathlib.RingTheory.Ideal.Operations.635_0.5qK551sG47yBciY

theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i}

Mathlib_RingTheory_Ideal_Operations

R✝ : Type u ι✝ : Type u_1 inst✝² : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u_2 inst✝¹ : CommRing R ι : Type u_3 inst✝ : Fintype ι I : ι → ℕ hI : ∀ (i j : ι), i ≠ j → Nat.Coprime (I i) (I j) ⊢ ⨅ i, span {↑(I i)} = span {↑(∏ i : ι, I i)}

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by

rw [iInf_span_singleton, Nat.cast_prod]

theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by

Mathlib.RingTheory.Ideal.Operations.642_0.5qK551sG47yBciY

theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)}

Mathlib_RingTheory_Ideal_Operations

R✝ : Type u ι✝ : Type u_1 inst✝² : CommSemiring R✝ I✝ J K L : Ideal R✝ R : Type u_2 inst✝¹ : CommRing R ι : Type u_3 inst✝ : Fintype ι I : ι → ℕ hI : ∀ (i j : ι), i ≠ j → Nat.Coprime (I i) (I j) ⊢ ∀ (i j : ι), i ≠ j → IsCoprime ↑(I i) ↑(I j)

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod]

exact fun i j h ↦ (hI i j h).cast

theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod]

Mathlib.RingTheory.Ideal.Operations.642_0.5qK551sG47yBciY

theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)}

Mathlib_RingTheory_Ideal_Operations

R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I J K L : Ideal R✝ R : Type u_2 inst✝ : CommSemiring R x y : R ⊢ span {x} ⊔ span {y} = ⊤ ↔ IsCoprime x y

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by

rw [eq_top_iff_one, Submodule.mem_sup]

theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by

Mathlib.RingTheory.Ideal.Operations.648_0.5qK551sG47yBciY

theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y

Mathlib_RingTheory_Ideal_Operations

R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I J K L : Ideal R✝ R : Type u_2 inst✝ : CommSemiring R x y : R ⊢ (∃ y_1 ∈ span {x}, ∃ z ∈ span {y}, y_1 + z = 1) ↔ IsCoprime x y

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup]

constructor

theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup]

Mathlib.RingTheory.Ideal.Operations.648_0.5qK551sG47yBciY

theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y

Mathlib_RingTheory_Ideal_Operations

case mp R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I J K L : Ideal R✝ R : Type u_2 inst✝ : CommSemiring R x y : R ⊢ (∃ y_1 ∈ span {x}, ∃ z ∈ span {y}, y_1 + z = 1) → IsCoprime x y

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor ·

rintro ⟨u, hu, v, hv, h1⟩

theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor ·

Mathlib.RingTheory.Ideal.Operations.648_0.5qK551sG47yBciY

theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y

Mathlib_RingTheory_Ideal_Operations

case mp.intro.intro.intro.intro R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I J K L : Ideal R✝ R : Type u_2 inst✝ : CommSemiring R x y u : R hu : u ∈ span {x} v : R hv : v ∈ span {y} h1 : u + v = 1 ⊢ IsCoprime x y

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩

rw [mem_span_singleton'] at hu hv

theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩

Mathlib.RingTheory.Ideal.Operations.648_0.5qK551sG47yBciY

theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y

Mathlib_RingTheory_Ideal_Operations

case mp.intro.intro.intro.intro R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I J K L : Ideal R✝ R : Type u_2 inst✝ : CommSemiring R x y u : R hu : ∃ a, a * x = u v : R hv : ∃ a, a * y = v h1 : u + v = 1 ⊢ IsCoprime x y

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv

rw [← hu.choose_spec, ← hv.choose_spec] at h1

theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv

Mathlib.RingTheory.Ideal.Operations.648_0.5qK551sG47yBciY

theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y

Mathlib_RingTheory_Ideal_Operations

case mp.intro.intro.intro.intro R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I J K L : Ideal R✝ R : Type u_2 inst✝ : CommSemiring R x y u : R hu : ∃ a, a * x = u v : R hv : ∃ a, a * y = v h1 : Exists.choose hu * x + Exists.choose hv * y = 1 ⊢ IsCoprime x y

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1

exact ⟨_, _, h1⟩

theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1

Mathlib.RingTheory.Ideal.Operations.648_0.5qK551sG47yBciY

theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y

Mathlib_RingTheory_Ideal_Operations

case mpr R✝ : Type u ι : Type u_1 inst✝¹ : CommSemiring R✝ I J K L : Ideal R✝ R : Type u_2 inst✝ : CommSemiring R x y : R ⊢ IsCoprime x y → ∃ y_1 ∈ span {x}, ∃ z ∈ span {y}, y_1 + z = 1

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ ·

exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩

theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ ·

Mathlib.RingTheory.Ideal.Operations.648_0.5qK551sG47yBciY

theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R s : Multiset (Ideal R) ⊢ Multiset.prod s ≤ Multiset.inf s

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by

classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih)

theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by

Mathlib.RingTheory.Ideal.Operations.664_0.5qK551sG47yBciY

theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R s : Multiset (Ideal R) ⊢ Multiset.prod s ≤ Multiset.inf s

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical

refine' s.induction_on _ _

theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical

Mathlib.RingTheory.Ideal.Operations.664_0.5qK551sG47yBciY

theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf

Mathlib_RingTheory_Ideal_Operations

case refine'_1 R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R s : Multiset (Ideal R) ⊢ Multiset.prod 0 ≤ Multiset.inf 0

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ ·

rw [Multiset.inf_zero]

theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ ·

Mathlib.RingTheory.Ideal.Operations.664_0.5qK551sG47yBciY

theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf

Mathlib_RingTheory_Ideal_Operations

case refine'_1 R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R s : Multiset (Ideal R) ⊢ Multiset.prod 0 ≤ ⊤

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero]

exact le_top

theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero]

Mathlib.RingTheory.Ideal.Operations.664_0.5qK551sG47yBciY

theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf

Mathlib_RingTheory_Ideal_Operations

case refine'_2 R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R s : Multiset (Ideal R) ⊢ ∀ ⦃a : Ideal R⦄ {s : Multiset (Ideal R)}, Multiset.prod s ≤ Multiset.inf s → Multiset.prod (a ::ₘ s) ≤ Multiset.inf (a ::ₘ s)

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top

intro a s ih

theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top

Mathlib.RingTheory.Ideal.Operations.664_0.5qK551sG47yBciY

theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf

Mathlib_RingTheory_Ideal_Operations

case refine'_2 R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R s✝ : Multiset (Ideal R) a : Ideal R s : Multiset (Ideal R) ih : Multiset.prod s ≤ Multiset.inf s ⊢ Multiset.prod (a ::ₘ s) ≤ Multiset.inf (a ::ₘ s)

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih

rw [Multiset.prod_cons, Multiset.inf_cons]

theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih

Mathlib.RingTheory.Ideal.Operations.664_0.5qK551sG47yBciY

theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf

Mathlib_RingTheory_Ideal_Operations

case refine'_2 R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R s✝ : Multiset (Ideal R) a : Ideal R s : Multiset (Ideal R) ih : Multiset.prod s ≤ Multiset.inf s ⊢ a * Multiset.prod s ≤ a ⊓ Multiset.inf s

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons]

exact le_trans mul_le_inf (inf_le_inf le_rfl ih)

theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons]

Mathlib.RingTheory.Ideal.Operations.664_0.5qK551sG47yBciY

theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R h : I ⊔ J = ⊤ i : R hi : i ∈ I ⊔ K ⊢ i ∈ I ⊔ J * K

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by

rw [eq_top_iff_one] at h

theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by

Mathlib.RingTheory.Ideal.Operations.686_0.5qK551sG47yBciY

theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R h : 1 ∈ I ⊔ J i : R hi : i ∈ I ⊔ K ⊢ i ∈ I ⊔ J * K

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h;

rw [Submodule.mem_sup] at h hi ⊢

theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h;

Mathlib.RingTheory.Ideal.Operations.686_0.5qK551sG47yBciY

theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R h : ∃ y ∈ I, ∃ z ∈ J, y + z = 1 i : R hi : ∃ y ∈ I, ∃ z ∈ K, y + z = i ⊢ ∃ y ∈ I, ∃ z ∈ J * K, y + z = i

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢

obtain ⟨i1, hi1, j, hj, h⟩ := h

theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢

Mathlib.RingTheory.Ideal.Operations.686_0.5qK551sG47yBciY

theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K

Mathlib_RingTheory_Ideal_Operations

case intro.intro.intro.intro R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R i : R hi : ∃ y ∈ I, ∃ z ∈ K, y + z = i i1 : R hi1 : i1 ∈ I j : R hj : j ∈ J h : i1 + j = 1 ⊢ ∃ y ∈ I, ∃ z ∈ J * K, y + z = i

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h;

obtain ⟨i', hi', k, hk, hi⟩ := hi

theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h;

Mathlib.RingTheory.Ideal.Operations.686_0.5qK551sG47yBciY

theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K

Mathlib_RingTheory_Ideal_Operations

case intro.intro.intro.intro.intro.intro.intro.intro R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R i i1 : R hi1 : i1 ∈ I j : R hj : j ∈ J h : i1 + j = 1 i' : R hi' : i' ∈ I k : R hk : k ∈ K hi : i' + k = i ⊢ ∃ y ∈ I, ∃ z ∈ J * K, y + z = i

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi

refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩

theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi

Mathlib.RingTheory.Ideal.Operations.686_0.5qK551sG47yBciY

theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K

Mathlib_RingTheory_Ideal_Operations

case intro.intro.intro.intro.intro.intro.intro.intro R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R i i1 : R hi1 : i1 ∈ I j : R hj : j ∈ J h : i1 + j = 1 i' : R hi' : i' ∈ I k : R hk : k ∈ K hi : i' + k = i ⊢ i' + i1 * k + j * k = i

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩

rw [add_assoc, ← add_mul, h, one_mul, hi]

theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩

Mathlib.RingTheory.Ideal.Operations.686_0.5qK551sG47yBciY

theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R h : I ⊔ K = ⊤ ⊢ I ⊔ J * K = I ⊔ J

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by

rw [mul_comm]

theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by

Mathlib.RingTheory.Ideal.Operations.694_0.5qK551sG47yBciY

theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R h : I ⊔ K = ⊤ ⊢ I ⊔ K * J = I ⊔ J

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm]

exact sup_mul_eq_of_coprime_left h

theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm]

Mathlib.RingTheory.Ideal.Operations.694_0.5qK551sG47yBciY

theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R h : I ⊔ J = ⊤ ⊢ I * K ⊔ J = K ⊔ J

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by

rw [sup_comm] at h

theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by

Mathlib.RingTheory.Ideal.Operations.699_0.5qK551sG47yBciY

theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R h : J ⊔ I = ⊤ ⊢ I * K ⊔ J = K ⊔ J

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h

rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm]

theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h

Mathlib.RingTheory.Ideal.Operations.699_0.5qK551sG47yBciY

theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R h : K ⊔ J = ⊤ ⊢ I * K ⊔ J = I ⊔ J

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by

rw [sup_comm] at h

theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by

Mathlib.RingTheory.Ideal.Operations.704_0.5qK551sG47yBciY

theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R h : J ⊔ K = ⊤ ⊢ I * K ⊔ J = I ⊔ J

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h

rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm]

theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h

Mathlib.RingTheory.Ideal.Operations.704_0.5qK551sG47yBciY

theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝ : CommSemiring R I J✝ K L : Ideal R s : Finset ι J : ι → Ideal R h : ∀ i ∈ s, I ⊔ J i = ⊤ ⊢ (fun J => I ⊔ J = ⊤) 1

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by

simp_rw [one_eq_top, sup_top_eq]

theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by

Mathlib.RingTheory.Ideal.Operations.709_0.5qK551sG47yBciY

theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R n : ℕ h : I ⊔ J = ⊤ ⊢ I ⊔ J ^ n = ⊤

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by

rw [← Finset.card_range n, ← Finset.prod_const]

theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by

Mathlib.RingTheory.Ideal.Operations.733_0.5qK551sG47yBciY

theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R n : ℕ h : I ⊔ J = ⊤ ⊢ I ⊔ ∏ _x in Finset.range n, J = ⊤

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const]

exact sup_prod_eq_top fun _ _ => h

theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const]

Mathlib.RingTheory.Ideal.Operations.733_0.5qK551sG47yBciY

theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R n : ℕ h : I ⊔ J = ⊤ ⊢ I ^ n ⊔ J = ⊤

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by

rw [← Finset.card_range n, ← Finset.prod_const]

theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by

Mathlib.RingTheory.Ideal.Operations.738_0.5qK551sG47yBciY

theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R n : ℕ h : I ⊔ J = ⊤ ⊢ (∏ _x in Finset.range n, I) ⊔ J = ⊤

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const]

exact prod_sup_eq_top fun _ _ => h

theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const]

Mathlib.RingTheory.Ideal.Operations.738_0.5qK551sG47yBciY

theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R ⊢ I * ⊥ = ⊥

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by

simp

theorem mul_bot : I * ⊥ = ⊥ := by

Mathlib.RingTheory.Ideal.Operations.750_0.5qK551sG47yBciY

theorem mul_bot : I * ⊥ = ⊥

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R ⊢ ⊥ * I = ⊥

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by

simp

theorem bot_mul : ⊥ * I = ⊥ := by

Mathlib.RingTheory.Ideal.Operations.754_0.5qK551sG47yBciY

theorem bot_mul : ⊥ * I = ⊥

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R m n : ℕ h : m ≤ n ⊢ I ^ n ≤ I ^ m

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by

cases' Nat.exists_eq_add_of_le h with k hk

theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by

Mathlib.RingTheory.Ideal.Operations.793_0.5qK551sG47yBciY

theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m

Mathlib_RingTheory_Ideal_Operations

case intro R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R m n : ℕ h : m ≤ n k : ℕ hk : n = m + k ⊢ I ^ n ≤ I ^ m

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk

rw [hk, pow_add]

theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk

Mathlib.RingTheory.Ideal.Operations.793_0.5qK551sG47yBciY

theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m

Mathlib_RingTheory_Ideal_Operations

case intro R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R m n : ℕ h : m ≤ n k : ℕ hk : n = m + k ⊢ I ^ m * I ^ k ≤ I ^ m

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add]

exact le_trans mul_le_inf inf_le_left

theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add]

Mathlib.RingTheory.Ideal.Operations.793_0.5qK551sG47yBciY

theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝ : CommSemiring R I✝ J✝ K L I J : Ideal R e : I ≤ J n : ℕ ⊢ I ^ n ≤ J ^ n

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by

induction' n with _ hn

theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by

Mathlib.RingTheory.Ideal.Operations.805_0.5qK551sG47yBciY

theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n

Mathlib_RingTheory_Ideal_Operations

case zero R : Type u ι : Type u_1 inst✝ : CommSemiring R I✝ J✝ K L I J : Ideal R e : I ≤ J ⊢ I ^ Nat.zero ≤ J ^ Nat.zero

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn ·

rw [pow_zero, pow_zero]

theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn ·

Mathlib.RingTheory.Ideal.Operations.805_0.5qK551sG47yBciY

theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n

Mathlib_RingTheory_Ideal_Operations

case succ R : Type u ι : Type u_1 inst✝ : CommSemiring R I✝ J✝ K L I J : Ideal R e : I ≤ J n✝ : ℕ hn : I ^ n✝ ≤ J ^ n✝ ⊢ I ^ Nat.succ n✝ ≤ J ^ Nat.succ n✝

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] ·

rw [pow_succ, pow_succ]

theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] ·

Mathlib.RingTheory.Ideal.Operations.805_0.5qK551sG47yBciY

theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n

Mathlib_RingTheory_Ideal_Operations

case succ R : Type u ι : Type u_1 inst✝ : CommSemiring R I✝ J✝ K L I J : Ideal R e : I ≤ J n✝ : ℕ hn : I ^ n✝ ≤ J ^ n✝ ⊢ I * I ^ n✝ ≤ J * J ^ n✝

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ]

exact Ideal.mul_mono e hn

theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ]

Mathlib.RingTheory.Ideal.Operations.805_0.5qK551sG47yBciY

theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n

Mathlib_RingTheory_Ideal_Operations

R✝ : Type u ι : Type u_1 inst✝² : CommSemiring R✝ I✝ J✝ K L : Ideal R✝ R : Type u_2 inst✝¹ : CommSemiring R inst✝ : NoZeroDivisors R I J : Ideal R h : I = ⊥ ∨ J = ⊥ ⊢ I * J = ⊥

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by

cases' h with h h

theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by

Mathlib.RingTheory.Ideal.Operations.812_0.5qK551sG47yBciY

theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥

Mathlib_RingTheory_Ideal_Operations

case inl R✝ : Type u ι : Type u_1 inst✝² : CommSemiring R✝ I✝ J✝ K L : Ideal R✝ R : Type u_2 inst✝¹ : CommSemiring R inst✝ : NoZeroDivisors R I J : Ideal R h : I = ⊥ ⊢ I * J = ⊥

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;>

rw [← Ideal.mul_bot, h, Ideal.mul_comm]

theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;>

Mathlib.RingTheory.Ideal.Operations.812_0.5qK551sG47yBciY

theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥

Mathlib_RingTheory_Ideal_Operations

case inr R✝ : Type u ι : Type u_1 inst✝² : CommSemiring R✝ I✝ J✝ K L : Ideal R✝ R : Type u_2 inst✝¹ : CommSemiring R inst✝ : NoZeroDivisors R I J : Ideal R h : J = ⊥ ⊢ I * J = ⊥

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;>

rw [← Ideal.mul_bot, h, Ideal.mul_comm]

theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;>

Mathlib.RingTheory.Ideal.Operations.812_0.5qK551sG47yBciY

theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥

Mathlib_RingTheory_Ideal_Operations

R✝ : Type u ι : Type u_1 inst✝² : CommSemiring R✝ I J K L : Ideal R✝ R : Type u_2 inst✝¹ : CommRing R inst✝ : IsDomain R s : Multiset (Ideal R) ⊢ Multiset.prod s = ⊥ ↔ ∃ I ∈ s, I = ⊥

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by

rw [bot_eq_zero, prod_zero_iff_exists_zero]

/-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by

Mathlib.RingTheory.Ideal.Operations.825_0.5qK551sG47yBciY

/-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥

Mathlib_RingTheory_Ideal_Operations

R✝ : Type u ι : Type u_1 inst✝² : CommSemiring R✝ I J K L : Ideal R✝ R : Type u_2 inst✝¹ : CommRing R inst✝ : IsDomain R s : Multiset (Ideal R) ⊢ (∃ r, ∃ (_ : r ∈ s), r = 0) ↔ ∃ I ∈ s, I = 0

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero]

simp

/-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero]

Mathlib.RingTheory.Ideal.Operations.825_0.5qK551sG47yBciY

/-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R w x y z : R ⊢ span {w, x} * span {y, z} = span {w * y, w * z, x * y, x * z}

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by

simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc]

theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by

Mathlib.RingTheory.Ideal.Operations.832_0.5qK551sG47yBciY

theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z}

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R ⊢ IsCoprime I J ↔ Codisjoint I J

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by

rw [IsCoprime, codisjoint_iff]

theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by

Mathlib.RingTheory.Ideal.Operations.837_0.5qK551sG47yBciY

theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R ⊢ (∃ a b, a * I + b * J = 1) ↔ I ⊔ J = ⊤

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff]

constructor

theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff]

Mathlib.RingTheory.Ideal.Operations.837_0.5qK551sG47yBciY

theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J

Mathlib_RingTheory_Ideal_Operations

case mp R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R ⊢ (∃ a b, a * I + b * J = 1) → I ⊔ J = ⊤

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor ·

rintro ⟨x, y, hxy⟩

theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor ·

Mathlib.RingTheory.Ideal.Operations.837_0.5qK551sG47yBciY

theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J

Mathlib_RingTheory_Ideal_Operations

case mp.intro.intro R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L x y : Ideal R hxy : x * I + y * J = 1 ⊢ I ⊔ J = ⊤

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩

rw [eq_top_iff_one]

theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩

Mathlib.RingTheory.Ideal.Operations.837_0.5qK551sG47yBciY

theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J

Mathlib_RingTheory_Ideal_Operations

case mp.intro.intro R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L x y : Ideal R hxy : x * I + y * J = 1 ⊢ 1 ∈ I ⊔ J

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one]

apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right))

theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one]

Mathlib.RingTheory.Ideal.Operations.837_0.5qK551sG47yBciY

theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J

Mathlib_RingTheory_Ideal_Operations

case mp.intro.intro.a R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L x y : Ideal R hxy : x * I + y * J = 1 ⊢ 1 ∈ x * I + y * J

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right))

rw [hxy]

theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right))

Mathlib.RingTheory.Ideal.Operations.837_0.5qK551sG47yBciY

theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J

Mathlib_RingTheory_Ideal_Operations

case mp.intro.intro.a R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L x y : Ideal R hxy : x * I + y * J = 1 ⊢ 1 ∈ 1

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy]

simp only [one_eq_top, Submodule.mem_top]

theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy]

Mathlib.RingTheory.Ideal.Operations.837_0.5qK551sG47yBciY

theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J

Mathlib_RingTheory_Ideal_Operations

case mpr R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R ⊢ I ⊔ J = ⊤ → ∃ a b, a * I + b * J = 1

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] ·

intro h

theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] ·

Mathlib.RingTheory.Ideal.Operations.837_0.5qK551sG47yBciY

theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J

Mathlib_RingTheory_Ideal_Operations

case mpr R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R h : I ⊔ J = ⊤ ⊢ ∃ a b, a * I + b * J = 1

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h

refine' ⟨1, 1, _⟩

theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h

Mathlib.RingTheory.Ideal.Operations.837_0.5qK551sG47yBciY

theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J

Mathlib_RingTheory_Ideal_Operations

case mpr R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R h : I ⊔ J = ⊤ ⊢ 1 * I + 1 * J = 1

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩

simpa only [one_eq_top, top_mul, Submodule.add_eq_sup]

theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩

Mathlib.RingTheory.Ideal.Operations.837_0.5qK551sG47yBciY

theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R ⊢ IsCoprime I J ↔ I + J = 1

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by

rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top]

theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by

Mathlib.RingTheory.Ideal.Operations.850_0.5qK551sG47yBciY

theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R ⊢ IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by

rw [← add_eq_one_iff, isCoprime_iff_add]

theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by

Mathlib.RingTheory.Ideal.Operations.853_0.5qK551sG47yBciY

theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R ⊢ IsCoprime I J ↔ I ⊔ J = ⊤

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by

rw [isCoprime_iff_codisjoint, codisjoint_iff]

theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by

Mathlib.RingTheory.Ideal.Operations.856_0.5qK551sG47yBciY

theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R ⊢ TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤]

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by

rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq]

open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by

Mathlib.RingTheory.Ideal.Operations.859_0.5qK551sG47yBciY

open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤]

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R ⊢ TFAE [IsCoprime I J, IsCoprime I J, IsCoprime I J, IsCoprime I J, IsCoprime I J]

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq]

simp

open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq]

Mathlib.RingTheory.Ideal.Operations.859_0.5qK551sG47yBciY

open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤]

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R x y : R ⊢ IsCoprime (span {x}) (span {y}) ↔ IsCoprime x y

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by

simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton]

theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by

Mathlib.RingTheory.Ideal.Operations.876_0.5qK551sG47yBciY

theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y

Mathlib_RingTheory_Ideal_Operations

R : Type u ι : Type u_1 inst✝ : CommSemiring R I J K L : Ideal R x y : R ⊢ (∃ a b, y ∣ b ∧ a * x + b = 1) ↔ IsCoprime x y

/- Copyright (c) 2018 Kenny Lau. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Kenny Lau -/ import Mathlib.Algebra.Algebra.Operations import Mathlib.Algebra.Ring.Equiv import Mathlib.Data.Nat.Choose.Sum import Mathlib.LinearAlgebra.Basis.Bilinear import Mathlib.RingTheory.Coprime.Lemmas import Mathlib.RingTheory.Ideal.Basic import Mathlib.Algebra.GroupWithZero.NonZeroDivisors #align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74" /-! # More operations on modules and ideals -/ universe u v w x open BigOperators Pointwise namespace Submodule variable {R : Type u} {M : Type v} {F : Type*} {G : Type*} section CommSemiring variable [CommSemiring R] [AddCommMonoid M] [Module R M] open Pointwise instance hasSMul' : SMul (Ideal R) (Submodule R M) := ⟨Submodule.map₂ (LinearMap.lsmul R M)⟩ #align submodule.has_smul' Submodule.hasSMul' /-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to apply. -/ protected theorem _root_.Ideal.smul_eq_mul (I J : Ideal R) : I • J = I * J := rfl #align ideal.smul_eq_mul Ideal.smul_eq_mul /-- `N.annihilator` is the ideal of all elements `r : R` such that `r • N = 0`. -/ def annihilator (N : Submodule R M) : Ideal R := LinearMap.ker (LinearMap.lsmul R N) #align submodule.annihilator Submodule.annihilator variable {I J : Ideal R} {N P : Submodule R M} theorem mem_annihilator {r} : r ∈ N.annihilator ↔ ∀ n ∈ N, r • n = (0 : M) := ⟨fun hr n hn => congr_arg Subtype.val (LinearMap.ext_iff.1 (LinearMap.mem_ker.1 hr) ⟨n, hn⟩), fun h => LinearMap.mem_ker.2 <| LinearMap.ext fun n => Subtype.eq <| h n.1 n.2⟩ #align submodule.mem_annihilator Submodule.mem_annihilator theorem mem_annihilator' {r} : r ∈ N.annihilator ↔ N ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) ⊥ := mem_annihilator.trans ⟨fun H n hn => (mem_bot R).2 <| H n hn, fun H _ hn => (mem_bot R).1 <| H hn⟩ #align submodule.mem_annihilator' Submodule.mem_annihilator' theorem mem_annihilator_span (s : Set M) (r : R) : r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by rw [Submodule.mem_annihilator] constructor · intro h n exact h _ (Submodule.subset_span n.prop) · intro h n hn refine Submodule.span_induction hn ?_ ?_ ?_ ?_ · intro x hx exact h ⟨x, hx⟩ · exact smul_zero _ · intro x y hx hy rw [smul_add, hx, hy, zero_add] · intro a x hx rw [smul_comm, hx, smul_zero] #align submodule.mem_annihilator_span Submodule.mem_annihilator_span theorem mem_annihilator_span_singleton (g : M) (r : R) : r ∈ (Submodule.span R ({g} : Set M)).annihilator ↔ r • g = 0 := by simp [mem_annihilator_span] #align submodule.mem_annihilator_span_singleton Submodule.mem_annihilator_span_singleton theorem annihilator_bot : (⊥ : Submodule R M).annihilator = ⊤ := (Ideal.eq_top_iff_one _).2 <| mem_annihilator'.2 bot_le #align submodule.annihilator_bot Submodule.annihilator_bot theorem annihilator_eq_top_iff : N.annihilator = ⊤ ↔ N = ⊥ := ⟨fun H => eq_bot_iff.2 fun (n : M) hn => (mem_bot R).2 <| one_smul R n ▸ mem_annihilator.1 ((Ideal.eq_top_iff_one _).1 H) n hn, fun H => H.symm ▸ annihilator_bot⟩ #align submodule.annihilator_eq_top_iff Submodule.annihilator_eq_top_iff theorem annihilator_mono (h : N ≤ P) : P.annihilator ≤ N.annihilator := fun _ hrp => mem_annihilator.2 fun n hn => mem_annihilator.1 hrp n <| h hn #align submodule.annihilator_mono Submodule.annihilator_mono theorem annihilator_iSup (ι : Sort w) (f : ι → Submodule R M) : annihilator (⨆ i, f i) = ⨅ i, annihilator (f i) := le_antisymm (le_iInf fun _ => annihilator_mono <| le_iSup _ _) fun _ H => mem_annihilator'.2 <| iSup_le fun i => have := (mem_iInf _).1 H i mem_annihilator'.1 this #align submodule.annihilator_supr Submodule.annihilator_iSup theorem smul_mem_smul {r} {n} (hr : r ∈ I) (hn : n ∈ N) : r • n ∈ I • N := apply_mem_map₂ _ hr hn #align submodule.smul_mem_smul Submodule.smul_mem_smul theorem smul_le {P : Submodule R M} : I • N ≤ P ↔ ∀ r ∈ I, ∀ n ∈ N, r • n ∈ P := map₂_le #align submodule.smul_le Submodule.smul_le @[elab_as_elim] theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (Hb : ∀ r ∈ I, ∀ n ∈ N, p (r • n)) (H1 : ∀ x y, p x → p y → p (x + y)) : p x := by have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem refine Submodule.iSup_induction (x := x) _ H ?_ H0 H1 rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩ rw [← hj'] exact Hb _ hi _ hj #align submodule.smul_induction_on Submodule.smul_induction_on /-- Dependent version of `Submodule.smul_induction_on`. -/ @[elab_as_elim] theorem smul_induction_on' {x : M} (hx : x ∈ I • N) {p : ∀ x, x ∈ I • N → Prop} (Hb : ∀ (r : R) (hr : r ∈ I) (n : M) (hn : n ∈ N), p (r • n) (smul_mem_smul hr hn)) (H1 : ∀ x hx y hy, p x hx → p y hy → p (x + y) (Submodule.add_mem _ ‹_› ‹_›)) : p x hx := by refine' Exists.elim _ fun (h : x ∈ I • N) (H : p x h) => H exact smul_induction_on hx (fun a ha x hx => ⟨_, Hb _ ha _ hx⟩) fun x y ⟨_, hx⟩ ⟨_, hy⟩ => ⟨_, H1 _ _ _ _ hx hy⟩ #align submodule.smul_induction_on' Submodule.smul_induction_on' theorem mem_smul_span_singleton {I : Ideal R} {m : M} {x : M} : x ∈ I • span R ({m} : Set M) ↔ ∃ y ∈ I, y • m = x := ⟨fun hx => smul_induction_on hx (fun r hri n hnm => let ⟨s, hs⟩ := mem_span_singleton.1 hnm ⟨r * s, I.mul_mem_right _ hri, hs ▸ mul_smul r s m⟩) fun m1 m2 ⟨y1, hyi1, hy1⟩ ⟨y2, hyi2, hy2⟩ => ⟨y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]⟩, fun ⟨y, hyi, hy⟩ => hy ▸ smul_mem_smul hyi (subset_span <| Set.mem_singleton m)⟩ #align submodule.mem_smul_span_singleton Submodule.mem_smul_span_singleton theorem smul_le_right : I • N ≤ N := smul_le.2 fun r _ _ => N.smul_mem r #align submodule.smul_le_right Submodule.smul_le_right theorem smul_mono (hij : I ≤ J) (hnp : N ≤ P) : I • N ≤ J • P := map₂_le_map₂ hij hnp #align submodule.smul_mono Submodule.smul_mono theorem smul_mono_left (h : I ≤ J) : I • N ≤ J • N := map₂_le_map₂_left h #align submodule.smul_mono_left Submodule.smul_mono_left theorem smul_mono_right (h : N ≤ P) : I • N ≤ I • P := map₂_le_map₂_right h #align submodule.smul_mono_right Submodule.smul_mono_right theorem map_le_smul_top (I : Ideal R) (f : R →ₗ[R] M) : Submodule.map f I ≤ I • (⊤ : Submodule R M) := by rintro _ ⟨y, hy, rfl⟩ rw [← mul_one y, ← smul_eq_mul, f.map_smul] exact smul_mem_smul hy mem_top #align submodule.map_le_smul_top Submodule.map_le_smul_top @[simp] theorem annihilator_smul (N : Submodule R M) : annihilator N • N = ⊥ := eq_bot_iff.2 (smul_le.2 fun _ => mem_annihilator.1) #align submodule.annihilator_smul Submodule.annihilator_smul @[simp] theorem annihilator_mul (I : Ideal R) : annihilator I * I = ⊥ := annihilator_smul I #align submodule.annihilator_mul Submodule.annihilator_mul @[simp] theorem mul_annihilator (I : Ideal R) : I * annihilator I = ⊥ := by rw [mul_comm, annihilator_mul] #align submodule.mul_annihilator Submodule.mul_annihilator variable (I J N P) @[simp] theorem smul_bot : I • (⊥ : Submodule R M) = ⊥ := map₂_bot_right _ _ #align submodule.smul_bot Submodule.smul_bot @[simp] theorem bot_smul : (⊥ : Ideal R) • N = ⊥ := map₂_bot_left _ _ #align submodule.bot_smul Submodule.bot_smul @[simp] theorem top_smul : (⊤ : Ideal R) • N = N := le_antisymm smul_le_right fun r hri => one_smul R r ▸ smul_mem_smul mem_top hri #align submodule.top_smul Submodule.top_smul theorem smul_sup : I • (N ⊔ P) = I • N ⊔ I • P := map₂_sup_right _ _ _ _ #align submodule.smul_sup Submodule.smul_sup theorem sup_smul : (I ⊔ J) • N = I • N ⊔ J • N := map₂_sup_left _ _ _ _ #align submodule.sup_smul Submodule.sup_smul protected theorem smul_assoc : (I • J) • N = I • J • N := le_antisymm (smul_le.2 fun _ hrsij t htn => smul_induction_on hrsij (fun r hr s hs => (@smul_eq_mul R _ r s).symm ▸ smul_smul r s t ▸ smul_mem_smul hr (smul_mem_smul hs htn)) fun x y => (add_smul x y t).symm ▸ Submodule.add_mem _) (smul_le.2 fun r hr _ hsn => suffices J • N ≤ Submodule.comap (r • (LinearMap.id : M →ₗ[R] M)) ((I • J) • N) from this hsn smul_le.2 fun s hs n hn => show r • s • n ∈ (I • J) • N from mul_smul r s n ▸ smul_mem_smul (smul_mem_smul hr hs) hn) #align submodule.smul_assoc Submodule.smul_assoc theorem smul_inf_le (M₁ M₂ : Submodule R M) : I • (M₁ ⊓ M₂) ≤ I • M₁ ⊓ I • M₂ := le_inf (Submodule.smul_mono_right inf_le_left) (Submodule.smul_mono_right inf_le_right) #align submodule.smul_inf_le Submodule.smul_inf_le theorem smul_iSup {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iSup t = ⨆ i, I • t i := map₂_iSup_right _ _ _ #align submodule.smul_supr Submodule.smul_iSup theorem smul_iInf_le {ι : Sort*} {I : Ideal R} {t : ι → Submodule R M} : I • iInf t ≤ ⨅ i, I • t i := le_iInf fun _ => smul_mono_right (iInf_le _ _) #align submodule.smul_infi_le Submodule.smul_iInf_le variable (S : Set R) (T : Set M) theorem span_smul_span : Ideal.span S • span R T = span R (⋃ (s ∈ S) (t ∈ T), {s • t}) := (map₂_span_span _ _ _ _).trans <| congr_arg _ <| Set.image2_eq_iUnion _ _ _ #align submodule.span_smul_span Submodule.span_smul_span theorem ideal_span_singleton_smul (r : R) (N : Submodule R M) : (Ideal.span {r} : Ideal R) • N = r • N := by have : span R (⋃ (t : M) (_ : t ∈ N), {r • t}) = r • N := by convert span_eq (r • N) exact (Set.image_eq_iUnion _ (N : Set M)).symm conv_lhs => rw [← span_eq N, span_smul_span] simpa #align submodule.ideal_span_singleton_smul Submodule.ideal_span_singleton_smul theorem mem_of_span_top_of_smul_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, (r : R) • x ∈ M') : x ∈ M' := by suffices (⊤ : Ideal R) • span R ({x} : Set M) ≤ M' by rw [top_smul] at this exact this (subset_span (Set.mem_singleton x)) rw [← hs, span_smul_span, span_le] simpa using H #align submodule.mem_of_span_top_of_smul_mem Submodule.mem_of_span_top_of_smul_mem /-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a submodule `M'` of `x`, we only need to show that `r ^ n • x ∈ M'` for some `n` for each `r : s`. -/ theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤) (x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩ choose n₁ n₂ using H let N := s'.attach.sup n₁ have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N apply M'.mem_of_span_top_of_smul_mem _ hs' rintro ⟨_, r, hr, rfl⟩ convert M'.smul_mem (r ^ (N - n₁ ⟨r, hr⟩)) (n₂ ⟨r, hr⟩) using 1 simp only [Subtype.coe_mk, smul_smul, ← pow_add] rw [tsub_add_cancel_of_le (Finset.le_sup (s'.mem_attach _) : n₁ ⟨r, hr⟩ ≤ N)] #align submodule.mem_of_span_eq_top_of_smul_pow_mem Submodule.mem_of_span_eq_top_of_smul_pow_mem variable {M' : Type w} [AddCommMonoid M'] [Module R M'] theorem map_smul'' (f : M →ₗ[R] M') : (I • N).map f = I • N.map f := le_antisymm (map_le_iff_le_comap.2 <| smul_le.2 fun r hr n hn => show f (r • n) ∈ I • N.map f from (f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <| smul_le.2 fun r hr _ hn => let ⟨p, hp, hfp⟩ := mem_map.1 hn hfp ▸ f.map_smul r p ▸ mem_map_of_mem (smul_mem_smul hr hp) #align submodule.map_smul'' Submodule.map_smul'' variable {I} theorem mem_smul_span {s : Set M} {x : M} : x ∈ I • Submodule.span R s ↔ x ∈ Submodule.span R (⋃ (a ∈ I) (b ∈ s), ({a • b} : Set M)) := by rw [← I.span_eq, Submodule.span_smul_span, I.span_eq] rfl #align submodule.mem_smul_span Submodule.mem_smul_span variable (I) /-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`, then we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/ theorem mem_ideal_smul_span_iff_exists_sum {ι : Type*} (f : ι → M) (x : M) : x ∈ I • span R (Set.range f) ↔ ∃ (a : ι →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by constructor; swap · rintro ⟨a, ha, rfl⟩ exact Submodule.sum_mem _ fun c _ => smul_mem_smul (ha c) <| subset_span <| Set.mem_range_self _ refine' fun hx => span_induction (mem_smul_span.mp hx) _ _ _ _ · simp only [Set.mem_iUnion, Set.mem_range, Set.mem_singleton_iff] rintro x ⟨y, hy, x, ⟨i, rfl⟩, rfl⟩ refine' ⟨Finsupp.single i y, fun j => _, _⟩ · letI := Classical.decEq ι rw [Finsupp.single_apply] split_ifs · assumption · exact I.zero_mem refine' @Finsupp.sum_single_index ι R M _ _ i _ (fun i y => y • f i) _ simp · exact ⟨0, fun _ => I.zero_mem, Finsupp.sum_zero_index⟩ · rintro x y ⟨ax, hax, rfl⟩ ⟨ay, hay, rfl⟩ refine' ⟨ax + ay, fun i => I.add_mem (hax i) (hay i), Finsupp.sum_add_index' _ _⟩ <;> intros <;> simp only [zero_smul, add_smul] · rintro c x ⟨a, ha, rfl⟩ refine' ⟨c • a, fun i => I.mul_mem_left c (ha i), _⟩ rw [Finsupp.sum_smul_index, Finsupp.smul_sum] <;> intros <;> simp only [zero_smul, mul_smul] #align submodule.mem_ideal_smul_span_iff_exists_sum Submodule.mem_ideal_smul_span_iff_exists_sum theorem mem_ideal_smul_span_iff_exists_sum' {ι : Type*} (s : Set ι) (f : ι → M) (x : M) : x ∈ I • span R (f '' s) ↔ ∃ (a : s →₀ R) (_ : ∀ i, a i ∈ I), (a.sum fun i c => c • f i) = x := by rw [← Submodule.mem_ideal_smul_span_iff_exists_sum, ← Set.image_eq_range] #align submodule.mem_ideal_smul_span_iff_exists_sum' Submodule.mem_ideal_smul_span_iff_exists_sum' theorem mem_smul_top_iff (N : Submodule R M) (x : N) : x ∈ I • (⊤ : Submodule R N) ↔ (x : M) ∈ I • N := by change _ ↔ N.subtype x ∈ I • N have : Submodule.map N.subtype (I • ⊤) = I • N := by rw [Submodule.map_smul'', Submodule.map_top, Submodule.range_subtype] rw [← this] exact (Function.Injective.mem_set_image N.injective_subtype).symm #align submodule.mem_smul_top_iff Submodule.mem_smul_top_iff @[simp] theorem smul_comap_le_comap_smul (f : M →ₗ[R] M') (S : Submodule R M') (I : Ideal R) : I • S.comap f ≤ (I • S).comap f := by refine' Submodule.smul_le.mpr fun r hr x hx => _ rw [Submodule.mem_comap] at hx ⊢ rw [f.map_smul] exact Submodule.smul_mem_smul hr hx #align submodule.smul_comap_le_comap_smul Submodule.smul_comap_le_comap_smul end CommSemiring section CommRing variable [CommRing R] [AddCommGroup M] [Module R M] variable {N N₁ N₂ P P₁ P₂ : Submodule R M} /-- `N.colon P` is the ideal of all elements `r : R` such that `r • P ⊆ N`. -/ def colon (N P : Submodule R M) : Ideal R := annihilator (P.map N.mkQ) #align submodule.colon Submodule.colon theorem mem_colon {r} : r ∈ N.colon P ↔ ∀ p ∈ P, r • p ∈ N := mem_annihilator.trans ⟨fun H p hp => (Quotient.mk_eq_zero N).1 (H (Quotient.mk p) (mem_map_of_mem hp)), fun H _ ⟨p, hp, hpm⟩ => hpm ▸ (N.mkQ.map_smul r p ▸ (Quotient.mk_eq_zero N).2 <| H p hp)⟩ #align submodule.mem_colon Submodule.mem_colon theorem mem_colon' {r} : r ∈ N.colon P ↔ P ≤ comap (r • (LinearMap.id : M →ₗ[R] M)) N := mem_colon #align submodule.mem_colon' Submodule.mem_colon' theorem colon_mono (hn : N₁ ≤ N₂) (hp : P₁ ≤ P₂) : N₁.colon P₂ ≤ N₂.colon P₁ := fun _ hrnp => mem_colon.2 fun p₁ hp₁ => hn <| mem_colon.1 hrnp p₁ <| hp hp₁ #align submodule.colon_mono Submodule.colon_mono theorem iInf_colon_iSup (ι₁ : Sort w) (f : ι₁ → Submodule R M) (ι₂ : Sort x) (g : ι₂ → Submodule R M) : (⨅ i, f i).colon (⨆ j, g j) = ⨅ (i) (j), (f i).colon (g j) := le_antisymm (le_iInf fun _ => le_iInf fun _ => colon_mono (iInf_le _ _) (le_iSup _ _)) fun _ H => mem_colon'.2 <| iSup_le fun j => map_le_iff_le_comap.1 <| le_iInf fun i => map_le_iff_le_comap.2 <| mem_colon'.1 <| have := (mem_iInf _).1 H i have := (mem_iInf _).1 this j this #align submodule.infi_colon_supr Submodule.iInf_colon_iSup @[simp] theorem mem_colon_singleton {N : Submodule R M} {x : M} {r : R} : r ∈ N.colon (Submodule.span R {x}) ↔ r • x ∈ N := calc r ∈ N.colon (Submodule.span R {x}) ↔ ∀ a : R, r • a • x ∈ N := by simp [Submodule.mem_colon, Submodule.mem_span_singleton] _ ↔ r • x ∈ N := by simp_rw [fun (a : R) ↦ smul_comm r a x]; exact SetLike.forall_smul_mem_iff #align submodule.mem_colon_singleton Submodule.mem_colon_singleton @[simp] theorem _root_.Ideal.mem_colon_singleton {I : Ideal R} {x r : R} : r ∈ I.colon (Ideal.span {x}) ↔ r * x ∈ I := by simp only [← Ideal.submodule_span_eq, Submodule.mem_colon_singleton, smul_eq_mul] #align ideal.mem_colon_singleton Ideal.mem_colon_singleton end CommRing end Submodule namespace Ideal section Add variable {R : Type u} [Semiring R] @[simp] theorem add_eq_sup {I J : Ideal R} : I + J = I ⊔ J := rfl #align ideal.add_eq_sup Ideal.add_eq_sup @[simp] theorem zero_eq_bot : (0 : Ideal R) = ⊥ := rfl #align ideal.zero_eq_bot Ideal.zero_eq_bot @[simp] theorem sum_eq_sup {ι : Type*} (s : Finset ι) (f : ι → Ideal R) : s.sum f = s.sup f := rfl #align ideal.sum_eq_sup Ideal.sum_eq_sup end Add section MulAndRadical variable {R : Type u} {ι : Type*} [CommSemiring R] variable {I J K L : Ideal R} instance : Mul (Ideal R) := ⟨(· • ·)⟩ @[simp] theorem one_eq_top : (1 : Ideal R) = ⊤ := by erw [Submodule.one_eq_range, LinearMap.range_id] #align ideal.one_eq_top Ideal.one_eq_top theorem add_eq_one_iff : I + J = 1 ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [one_eq_top, eq_top_iff_one, add_eq_sup, Submodule.mem_sup] theorem mul_mem_mul {r s} (hr : r ∈ I) (hs : s ∈ J) : r * s ∈ I * J := Submodule.smul_mem_smul hr hs #align ideal.mul_mem_mul Ideal.mul_mem_mul theorem mul_mem_mul_rev {r s} (hr : r ∈ I) (hs : s ∈ J) : s * r ∈ I * J := mul_comm r s ▸ mul_mem_mul hr hs #align ideal.mul_mem_mul_rev Ideal.mul_mem_mul_rev theorem pow_mem_pow {x : R} (hx : x ∈ I) (n : ℕ) : x ^ n ∈ I ^ n := Submodule.pow_mem_pow _ hx _ #align ideal.pow_mem_pow Ideal.pow_mem_pow theorem prod_mem_prod {ι : Type*} {s : Finset ι} {I : ι → Ideal R} {x : ι → R} : (∀ i ∈ s, x i ∈ I i) → (∏ i in s, x i) ∈ ∏ i in s, I i := by classical refine Finset.induction_on s ?_ ?_ · intro rw [Finset.prod_empty, Finset.prod_empty, one_eq_top] exact Submodule.mem_top · intro a s ha IH h rw [Finset.prod_insert ha, Finset.prod_insert ha] exact mul_mem_mul (h a <| Finset.mem_insert_self a s) (IH fun i hi => h i <| Finset.mem_insert_of_mem hi) #align ideal.prod_mem_prod Ideal.prod_mem_prod theorem mul_le : I * J ≤ K ↔ ∀ r ∈ I, ∀ s ∈ J, r * s ∈ K := Submodule.smul_le #align ideal.mul_le Ideal.mul_le theorem mul_le_left : I * J ≤ J := Ideal.mul_le.2 fun _ _ _ => J.mul_mem_left _ #align ideal.mul_le_left Ideal.mul_le_left theorem mul_le_right : I * J ≤ I := Ideal.mul_le.2 fun _ hr _ _ => I.mul_mem_right _ hr #align ideal.mul_le_right Ideal.mul_le_right @[simp] theorem sup_mul_right_self : I ⊔ I * J = I := sup_eq_left.2 Ideal.mul_le_right #align ideal.sup_mul_right_self Ideal.sup_mul_right_self @[simp] theorem sup_mul_left_self : I ⊔ J * I = I := sup_eq_left.2 Ideal.mul_le_left #align ideal.sup_mul_left_self Ideal.sup_mul_left_self @[simp] theorem mul_right_self_sup : I * J ⊔ I = I := sup_eq_right.2 Ideal.mul_le_right #align ideal.mul_right_self_sup Ideal.mul_right_self_sup @[simp] theorem mul_left_self_sup : J * I ⊔ I = I := sup_eq_right.2 Ideal.mul_le_left #align ideal.mul_left_self_sup Ideal.mul_left_self_sup variable (I J K) protected theorem mul_comm : I * J = J * I := le_antisymm (mul_le.2 fun _ hrI _ hsJ => mul_mem_mul_rev hsJ hrI) (mul_le.2 fun _ hrJ _ hsI => mul_mem_mul_rev hsI hrJ) #align ideal.mul_comm Ideal.mul_comm protected theorem mul_assoc : I * J * K = I * (J * K) := Submodule.smul_assoc I J K #align ideal.mul_assoc Ideal.mul_assoc theorem span_mul_span (S T : Set R) : span S * span T = span (⋃ (s ∈ S) (t ∈ T), {s * t}) := Submodule.span_smul_span S T #align ideal.span_mul_span Ideal.span_mul_span variable {I J K} theorem span_mul_span' (S T : Set R) : span S * span T = span (S * T) := by unfold span rw [Submodule.span_mul_span] #align ideal.span_mul_span' Ideal.span_mul_span' theorem span_singleton_mul_span_singleton (r s : R) : span {r} * span {s} = (span {r * s} : Ideal R) := by unfold span rw [Submodule.span_mul_span, Set.singleton_mul_singleton] #align ideal.span_singleton_mul_span_singleton Ideal.span_singleton_mul_span_singleton theorem span_singleton_pow (s : R) (n : ℕ) : span {s} ^ n = (span {s ^ n} : Ideal R) := by induction' n with n ih; · simp [Set.singleton_one] simp only [pow_succ, ih, span_singleton_mul_span_singleton] #align ideal.span_singleton_pow Ideal.span_singleton_pow theorem mem_mul_span_singleton {x y : R} {I : Ideal R} : x ∈ I * span {y} ↔ ∃ z ∈ I, z * y = x := Submodule.mem_smul_span_singleton #align ideal.mem_mul_span_singleton Ideal.mem_mul_span_singleton theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by simp only [mul_comm, mem_mul_span_singleton] #align ideal.mem_span_singleton_mul Ideal.mem_span_singleton_mul theorem le_span_singleton_mul_iff {x : R} {I J : Ideal R} : I ≤ span {x} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI := show (∀ {zI} (_ : zI ∈ I), zI ∈ span {x} * J) ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI by simp only [mem_span_singleton_mul] #align ideal.le_span_singleton_mul_iff Ideal.le_span_singleton_mul_iff theorem span_singleton_mul_le_iff {x : R} {I J : Ideal R} : span {x} * I ≤ J ↔ ∀ z ∈ I, x * z ∈ J := by simp only [mul_le, mem_span_singleton_mul, mem_span_singleton] constructor · intro h zI hzI exact h x (dvd_refl x) zI hzI · rintro h _ ⟨z, rfl⟩ zI hzI rw [mul_comm x z, mul_assoc] exact J.mul_mem_left _ (h zI hzI) #align ideal.span_singleton_mul_le_iff Ideal.span_singleton_mul_le_iff theorem span_singleton_mul_le_span_singleton_mul {x y : R} {I J : Ideal R} : span {x} * I ≤ span {y} * J ↔ ∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ := by simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_le_span_singleton_mul Ideal.span_singleton_mul_le_span_singleton_mul theorem span_singleton_mul_right_mono [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I ≤ span {x} * J ↔ I ≤ J := by simp_rw [span_singleton_mul_le_span_singleton_mul, mul_right_inj' hx, exists_eq_right', SetLike.le_def] #align ideal.span_singleton_mul_right_mono Ideal.span_singleton_mul_right_mono theorem span_singleton_mul_left_mono [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} ≤ J * span {x} ↔ I ≤ J := by simpa only [mul_comm I, mul_comm J] using span_singleton_mul_right_mono hx #align ideal.span_singleton_mul_left_mono Ideal.span_singleton_mul_left_mono theorem span_singleton_mul_right_inj [IsDomain R] {x : R} (hx : x ≠ 0) : span {x} * I = span {x} * J ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_right_mono hx] #align ideal.span_singleton_mul_right_inj Ideal.span_singleton_mul_right_inj theorem span_singleton_mul_left_inj [IsDomain R] {x : R} (hx : x ≠ 0) : I * span {x} = J * span {x} ↔ I = J := by simp only [le_antisymm_iff, span_singleton_mul_left_mono hx] #align ideal.span_singleton_mul_left_inj Ideal.span_singleton_mul_left_inj theorem span_singleton_mul_right_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective ((span {x} : Ideal R) * ·) := fun _ _ => (span_singleton_mul_right_inj hx).mp #align ideal.span_singleton_mul_right_injective Ideal.span_singleton_mul_right_injective theorem span_singleton_mul_left_injective [IsDomain R] {x : R} (hx : x ≠ 0) : Function.Injective fun I : Ideal R => I * span {x} := fun _ _ => (span_singleton_mul_left_inj hx).mp #align ideal.span_singleton_mul_left_injective Ideal.span_singleton_mul_left_injective theorem eq_span_singleton_mul {x : R} (I J : Ideal R) : I = span {x} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zJ = zI) ∧ ∀ z ∈ J, x * z ∈ I := by simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff] #align ideal.eq_span_singleton_mul Ideal.eq_span_singleton_mul theorem span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : Ideal R) : span {x} * I = span {y} * J ↔ (∀ zI ∈ I, ∃ zJ ∈ J, x * zI = y * zJ) ∧ ∀ zJ ∈ J, ∃ zI ∈ I, x * zI = y * zJ := by simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm] #align ideal.span_singleton_mul_eq_span_singleton_mul Ideal.span_singleton_mul_eq_span_singleton_mul theorem prod_span {ι : Type*} (s : Finset ι) (I : ι → Set R) : (∏ i in s, Ideal.span (I i)) = Ideal.span (∏ i in s, I i) := Submodule.prod_span s I #align ideal.prod_span Ideal.prod_span theorem prod_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) : (∏ i in s, Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := Submodule.prod_span_singleton s I #align ideal.prod_span_singleton Ideal.prod_span_singleton @[simp] theorem multiset_prod_span_singleton (m : Multiset R) : (m.map fun x => Ideal.span {x}).prod = Ideal.span ({Multiset.prod m} : Set R) := Multiset.induction_on m (by simp) fun a m ih => by simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton] #align ideal.multiset_prod_span_singleton Ideal.multiset_prod_span_singleton theorem finset_inf_span_singleton {ι : Type*} (s : Finset ι) (I : ι → R) (hI : Set.Pairwise (↑s) (IsCoprime on I)) : (s.inf fun i => Ideal.span ({I i} : Set R)) = Ideal.span {∏ i in s, I i} := by ext x simp only [Submodule.mem_finset_inf, Ideal.mem_span_singleton] exact ⟨Finset.prod_dvd_of_coprime hI, fun h i hi => (Finset.dvd_prod_of_mem _ hi).trans h⟩ #align ideal.finset_inf_span_singleton Ideal.finset_inf_span_singleton theorem iInf_span_singleton {ι : Type*} [Fintype ι] {I : ι → R} (hI : ∀ (i j) (_ : i ≠ j), IsCoprime (I i) (I j)) : ⨅ i, span ({I i} : Set R) = span {∏ i, I i} := by rw [← Finset.inf_univ_eq_iInf, finset_inf_span_singleton] rwa [Finset.coe_univ, Set.pairwise_univ] #align ideal.infi_span_singleton Ideal.iInf_span_singleton theorem iInf_span_singleton_natCast {R : Type*} [CommRing R] {ι : Type*} [Fintype ι] {I : ι → ℕ} (hI : ∀ (i j : ι), i ≠ j → (I i).Coprime (I j)) : ⨅ (i : ι), span {(I i : R)} = span {((∏ i : ι, I i : ℕ) : R)} := by rw [iInf_span_singleton, Nat.cast_prod] exact fun i j h ↦ (hI i j h).cast theorem sup_eq_top_iff_isCoprime {R : Type*} [CommSemiring R] (x y : R) : span ({x} : Set R) ⊔ span {y} = ⊤ ↔ IsCoprime x y := by rw [eq_top_iff_one, Submodule.mem_sup] constructor · rintro ⟨u, hu, v, hv, h1⟩ rw [mem_span_singleton'] at hu hv rw [← hu.choose_spec, ← hv.choose_spec] at h1 exact ⟨_, _, h1⟩ · exact fun ⟨u, v, h1⟩ => ⟨_, mem_span_singleton'.mpr ⟨_, rfl⟩, _, mem_span_singleton'.mpr ⟨_, rfl⟩, h1⟩ #align ideal.sup_eq_top_iff_is_coprime Ideal.sup_eq_top_iff_isCoprime theorem mul_le_inf : I * J ≤ I ⊓ J := mul_le.2 fun r hri s hsj => ⟨I.mul_mem_right s hri, J.mul_mem_left r hsj⟩ #align ideal.mul_le_inf Ideal.mul_le_inf theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod ≤ s.inf := by classical refine' s.induction_on _ _ · rw [Multiset.inf_zero] exact le_top intro a s ih rw [Multiset.prod_cons, Multiset.inf_cons] exact le_trans mul_le_inf (inf_le_inf le_rfl ih) #align ideal.multiset_prod_le_inf Ideal.multiset_prod_le_inf theorem prod_le_inf {s : Finset ι} {f : ι → Ideal R} : s.prod f ≤ s.inf f := multiset_prod_le_inf #align ideal.prod_le_inf Ideal.prod_le_inf theorem mul_eq_inf_of_coprime (h : I ⊔ J = ⊤) : I * J = I ⊓ J := le_antisymm mul_le_inf fun r ⟨hri, hrj⟩ => let ⟨s, hsi, t, htj, hst⟩ := Submodule.mem_sup.1 ((eq_top_iff_one _).1 h) mul_one r ▸ hst ▸ (mul_add r s t).symm ▸ Ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj) (mul_mem_mul hri htj) #align ideal.mul_eq_inf_of_coprime Ideal.mul_eq_inf_of_coprime theorem sup_mul_eq_of_coprime_left (h : I ⊔ J = ⊤) : I ⊔ J * K = I ⊔ K := le_antisymm (sup_le_sup_left mul_le_left _) fun i hi => by rw [eq_top_iff_one] at h; rw [Submodule.mem_sup] at h hi ⊢ obtain ⟨i1, hi1, j, hj, h⟩ := h; obtain ⟨i', hi', k, hk, hi⟩ := hi refine' ⟨_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _⟩ rw [add_assoc, ← add_mul, h, one_mul, hi] #align ideal.sup_mul_eq_of_coprime_left Ideal.sup_mul_eq_of_coprime_left theorem sup_mul_eq_of_coprime_right (h : I ⊔ K = ⊤) : I ⊔ J * K = I ⊔ J := by rw [mul_comm] exact sup_mul_eq_of_coprime_left h #align ideal.sup_mul_eq_of_coprime_right Ideal.sup_mul_eq_of_coprime_right theorem mul_sup_eq_of_coprime_left (h : I ⊔ J = ⊤) : I * K ⊔ J = K ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] #align ideal.mul_sup_eq_of_coprime_left Ideal.mul_sup_eq_of_coprime_left theorem mul_sup_eq_of_coprime_right (h : K ⊔ J = ⊤) : I * K ⊔ J = I ⊔ J := by rw [sup_comm] at h rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] #align ideal.mul_sup_eq_of_coprime_right Ideal.mul_sup_eq_of_coprime_right theorem sup_prod_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ∏ i in s, J i) = ⊤ := Finset.prod_induction _ (fun J => I ⊔ J = ⊤) (fun J K hJ hK => (sup_mul_eq_of_coprime_left hJ).trans hK) (by simp_rw [one_eq_top, sup_top_eq]) h #align ideal.sup_prod_eq_top Ideal.sup_prod_eq_top theorem sup_iInf_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → I ⊔ J i = ⊤) : (I ⊔ ⨅ i ∈ s, J i) = ⊤ := eq_top_iff.mpr <| le_of_eq_of_le (sup_prod_eq_top h).symm <| sup_le_sup_left (le_of_le_of_eq prod_le_inf <| Finset.inf_eq_iInf _ _) _ #align ideal.sup_infi_eq_top Ideal.sup_iInf_eq_top theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (∏ i in s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_prod_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.prod_sup_eq_top Ideal.prod_sup_eq_top theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) : (⨅ i ∈ s, J i) ⊔ I = ⊤ := sup_comm.trans (sup_iInf_eq_top fun i hi => sup_comm.trans <| h i hi) #align ideal.infi_sup_eq_top Ideal.iInf_sup_eq_top theorem sup_pow_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ⊔ J ^ n = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact sup_prod_eq_top fun _ _ => h #align ideal.sup_pow_eq_top Ideal.sup_pow_eq_top theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by rw [← Finset.card_range n, ← Finset.prod_const] exact prod_sup_eq_top fun _ _ => h #align ideal.pow_sup_eq_top Ideal.pow_sup_eq_top theorem pow_sup_pow_eq_top {m n : ℕ} (h : I ⊔ J = ⊤) : I ^ m ⊔ J ^ n = ⊤ := sup_pow_eq_top (pow_sup_eq_top h) #align ideal.pow_sup_pow_eq_top Ideal.pow_sup_pow_eq_top variable (I) -- @[simp] -- Porting note: simp can prove this theorem mul_bot : I * ⊥ = ⊥ := by simp #align ideal.mul_bot Ideal.mul_bot -- @[simp] -- Porting note: simp can prove this theorem bot_mul : ⊥ * I = ⊥ := by simp #align ideal.bot_mul Ideal.bot_mul @[simp] theorem mul_top : I * ⊤ = I := Ideal.mul_comm ⊤ I ▸ Submodule.top_smul I #align ideal.mul_top Ideal.mul_top @[simp] theorem top_mul : ⊤ * I = I := Submodule.top_smul I #align ideal.top_mul Ideal.top_mul variable {I} theorem mul_mono (hik : I ≤ K) (hjl : J ≤ L) : I * J ≤ K * L := Submodule.smul_mono hik hjl #align ideal.mul_mono Ideal.mul_mono theorem mul_mono_left (h : I ≤ J) : I * K ≤ J * K := Submodule.smul_mono_left h #align ideal.mul_mono_left Ideal.mul_mono_left theorem mul_mono_right (h : J ≤ K) : I * J ≤ I * K := Submodule.smul_mono_right h #align ideal.mul_mono_right Ideal.mul_mono_right variable (I J K) theorem mul_sup : I * (J ⊔ K) = I * J ⊔ I * K := Submodule.smul_sup I J K #align ideal.mul_sup Ideal.mul_sup theorem sup_mul : (I ⊔ J) * K = I * K ⊔ J * K := Submodule.sup_smul I J K #align ideal.sup_mul Ideal.sup_mul variable {I J K} theorem pow_le_pow_right {m n : ℕ} (h : m ≤ n) : I ^ n ≤ I ^ m := by cases' Nat.exists_eq_add_of_le h with k hk rw [hk, pow_add] exact le_trans mul_le_inf inf_le_left #align ideal.pow_le_pow_right Ideal.pow_le_pow_right theorem pow_le_self {n : ℕ} (hn : n ≠ 0) : I ^ n ≤ I := calc I ^ n ≤ I ^ 1 := pow_le_pow_right (Nat.pos_of_ne_zero hn) _ = I := pow_one _ #align ideal.pow_le_self Ideal.pow_le_self theorem pow_right_mono {I J : Ideal R} (e : I ≤ J) (n : ℕ) : I ^ n ≤ J ^ n := by induction' n with _ hn · rw [pow_zero, pow_zero] · rw [pow_succ, pow_succ] exact Ideal.mul_mono e hn #align ideal.pow_right_mono Ideal.pow_right_mono theorem mul_eq_bot {R : Type*} [CommSemiring R] [NoZeroDivisors R] {I J : Ideal R} : I * J = ⊥ ↔ I = ⊥ ∨ J = ⊥ := ⟨fun hij => or_iff_not_imp_left.mpr fun I_ne_bot => J.eq_bot_iff.mpr fun j hj => let ⟨i, hi, ne0⟩ := I.ne_bot_iff.mp I_ne_bot Or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0, fun h => by cases' h with h h <;> rw [← Ideal.mul_bot, h, Ideal.mul_comm]⟩ #align ideal.mul_eq_bot Ideal.mul_eq_bot instance {R : Type*} [CommSemiring R] [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1 /-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/ theorem prod_eq_bot {R : Type*} [CommRing R] [IsDomain R] {s : Multiset (Ideal R)} : s.prod = ⊥ ↔ ∃ I ∈ s, I = ⊥ := by rw [bot_eq_zero, prod_zero_iff_exists_zero] simp #align ideal.prod_eq_bot Ideal.prod_eq_bot theorem span_pair_mul_span_pair (w x y z : R) : (span {w, x} : Ideal R) * span {y, z} = span {w * y, w * z, x * y, x * z} := by simp_rw [span_insert, sup_mul, mul_sup, span_singleton_mul_span_singleton, sup_assoc] #align ideal.span_pair_mul_span_pair Ideal.span_pair_mul_span_pair theorem isCoprime_iff_codisjoint : IsCoprime I J ↔ Codisjoint I J := by rw [IsCoprime, codisjoint_iff] constructor · rintro ⟨x, y, hxy⟩ rw [eq_top_iff_one] apply (show x * I + y * J ≤ I ⊔ J from sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right)) rw [hxy] simp only [one_eq_top, Submodule.mem_top] · intro h refine' ⟨1, 1, _⟩ simpa only [one_eq_top, top_mul, Submodule.add_eq_sup] theorem isCoprime_iff_add : IsCoprime I J ↔ I + J = 1 := by rw [isCoprime_iff_codisjoint, codisjoint_iff, add_eq_sup, one_eq_top] theorem isCoprime_iff_exists : IsCoprime I J ↔ ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := by rw [← add_eq_one_iff, isCoprime_iff_add] theorem isCoprime_iff_sup_eq : IsCoprime I J ↔ I ⊔ J = ⊤ := by rw [isCoprime_iff_codisjoint, codisjoint_iff] open List in theorem isCoprime_tfae : TFAE [IsCoprime I J, Codisjoint I J, I + J = 1, ∃ i ∈ I, ∃ j ∈ J, i + j = 1, I ⊔ J = ⊤] := by rw [← isCoprime_iff_codisjoint, ← isCoprime_iff_add, ← isCoprime_iff_exists, ← isCoprime_iff_sup_eq] simp theorem _root_.IsCoprime.codisjoint (h : IsCoprime I J) : Codisjoint I J := isCoprime_iff_codisjoint.mp h theorem _root_.IsCoprime.add_eq (h : IsCoprime I J) : I + J = 1 := isCoprime_iff_add.mp h theorem _root_.IsCoprime.exists (h : IsCoprime I J) : ∃ i ∈ I, ∃ j ∈ J, i + j = 1 := isCoprime_iff_exists.mp h theorem _root_.IsCoprime.sup_eq (h : IsCoprime I J) : I ⊔ J = ⊤ := isCoprime_iff_sup_eq.mp h theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton]

constructor

theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y := by simp_rw [isCoprime_iff_codisjoint, codisjoint_iff, eq_top_iff_one, mem_span_singleton_sup, mem_span_singleton]

Mathlib.RingTheory.Ideal.Operations.876_0.5qK551sG47yBciY

theorem isCoprime_span_singleton_iff (x y : R) : IsCoprime (span <| singleton x) (span <| singleton y) ↔ IsCoprime x y

Mathlib_RingTheory_Ideal_Operations