File: Mathlib/RingTheory/Localization/Ideal.lean

full_name: _private.Mathlib.RingTheory.Localization.Ideal.0.IsLocalization.map_ideal
code: /-- Explicit characterization of the ideal given by `Ideal.map (algebraMap R S) I`.
In practice, this ideal differs only in that the carrier set is defined explicitly.
This definition is only meant to be used in proving `mem_map_algebraMap_iff`,
and any proof that needs to refer to the explicit carrier set should use that theorem. -/
private def map_ideal (I : Ideal R) : Ideal S 
goal: Ideal S

full_name: IsLocalization.mem_map_algebraMap_iff
code: theorem mem_map_algebraMap_iff {I : Ideal R} {z} : z ∈ Ideal.map (algebraMap R S) I ↔
    ∃ x : I × M, z * algebraMap R S x.2 = algebraMap R S x.1 
goal: z ∈ Ideal.map (algebraMap R S) I ↔ ∃ x, z * (algebraMap R S) ↑x.2 = (algebraMap R S) ↑x.1

full_name: IsLocalization.mk'_mem_map_algebraMap_iff
code: lemma mk'_mem_map_algebraMap_iff (I : Ideal R) (x : R) (s : M) :
    IsLocalization.mk' S x s ∈ I.map (algebraMap R S) ↔ ∃ s ∈ M, s * x ∈ I 
goal: IsLocalization.mk' S x s ∈ Ideal.map (algebraMap R S) I ↔ ∃ s ∈ M, s * x ∈ I

full_name: IsLocalization.map_comap
code: theorem map_comap [IsLocalization M S] (J : Ideal S) :
    Ideal.map (algebraMap R S) (Ideal.comap (algebraMap R S) J) = J 
goal: Ideal.map (algebraMap R S) (Ideal.comap (algebraMap R S) J) = J

full_name: IsLocalization.comap_map_of_isPrime_disjoint
code: theorem comap_map_of_isPrime_disjoint (I : Ideal R) (hI : I.IsPrime) (hM : Disjoint (M : Set R) I) :
    Ideal.comap (algebraMap R S) (Ideal.map (algebraMap R S) I) = I 
goal: Ideal.comap (algebraMap R S) (Ideal.map (algebraMap R S) I) = I

full_name: IsLocalization.orderEmbedding
code: /-- If `S` is the localization of `R` at a submonoid, the ordering of ideals of `S` is
embedded in the ordering of ideals of `R`. -/
def orderEmbedding : Ideal S ↪o Ideal R 
goal: Ideal S ↪o Ideal R

full_name: IsLocalization.isPrime_iff_isPrime_disjoint
code: /-- If `R` is a ring, then prime ideals in the localization at `M`
correspond to prime ideals in the original ring `R` that are disjoint from `M`.
This lemma gives the particular case for an ideal and its comap,
see `le_rel_iso_of_prime` for the more general relation isomorphism -/
theorem isPrime_iff_isPrime_disjoint (J : Ideal S) :
    J.IsPrime ↔
      (Ideal.comap (algebraMap R S) J).IsPrime ∧
        Disjoint (M : Set R) ↑(Ideal.comap (algebraMap R S) J) 
goal: J.IsPrime ↔ (Ideal.comap (algebraMap R S) J).IsPrime ∧ Disjoint ↑M ↑(Ideal.comap (algebraMap R S) J)

full_name: IsLocalization.isPrime_of_isPrime_disjoint
code: /-- If `R` is a ring, then prime ideals in the localization at `M`
correspond to prime ideals in the original ring `R` that are disjoint from `M`.
This lemma gives the particular case for an ideal and its map,
see `le_rel_iso_of_prime` for the more general relation isomorphism, and the reverse implication -/
theorem isPrime_of_isPrime_disjoint (I : Ideal R) (hp : I.IsPrime) (hd : Disjoint (M : Set R) ↑I) :
    (Ideal.map (algebraMap R S) I).IsPrime 
goal: (Ideal.map (algebraMap R S) I).IsPrime

full_name: IsLocalization.orderIsoOfPrime
code: /-- If `R` is a ring, then prime ideals in the localization at `M`
correspond to prime ideals in the original ring `R` that are disjoint from `M` -/
def orderIsoOfPrime :
    { p : Ideal S // p.IsPrime } ≃o { p : Ideal R // p.IsPrime ∧ Disjoint (M : Set R) ↑p } 
goal: { p // p.IsPrime } ≃o { p // p.IsPrime ∧ Disjoint ↑M ↑p }

full_name: IsLocalization.surjective_quotientMap_of_maximal_of_localization
code: /-- `quotientMap` applied to maximal ideals of a localization is `surjective`.
  The quotient by a maximal ideal is a field, so inverses to elements already exist,
  and the localization necessarily maps the equivalence class of the inverse in the localization -/
theorem surjective_quotientMap_of_maximal_of_localization {I : Ideal S} [I.IsPrime] {J : Ideal R}
    {H : J ≤ I.comap (algebraMap R S)} (hI : (I.comap (algebraMap R S)).IsMaximal) :
    Function.Surjective (Ideal.quotientMap I (algebraMap R S) H) 
goal: Function.Surjective ⇑(Ideal.quotientMap I (algebraMap R S) H)

full_name: IsLocalization.bot_lt_comap_prime
code: theorem bot_lt_comap_prime [IsDomain R] (hM : M ≤ R⁰) (p : Ideal S) [hpp : p.IsPrime]
    (hp0 : p ≠ ⊥) : ⊥ < Ideal.comap (algebraMap R S) p 
goal: ⊥ < Ideal.comap (algebraMap R S) p

full_name: IsLocalization.ideal_eq_iInf_comap_map_away
code: theorem ideal_eq_iInf_comap_map_away {S : Finset R} (hS : Ideal.span (α := R) S = ⊤) (I : Ideal R) :
    I = ⨅ f ∈ S, (I.map (algebraMap R (Localization.Away f))).comap
    (algebraMap R (Localization.Away f)) 
goal: I = ⨅ f ∈ S, Ideal.comap (algebraMap R (Localization.Away f)) (Ideal.map (algebraMap R (Localization.Away f)) I)