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import ring_theory.noetherian
import ring_theory.algebra_operations
local attribute [instance] classical.prop_decidable
namespace submodule
open algebra
variables {R : Type*} {A : Type*} [comm_ring R] [comm_ring A] [algebra R A]
local attribute [instance] set.pointwise_mul_semiring
local attribute [instance] set.singleton.is_monoid_hom
set_option class.instance_max_depth 80
lemma smul_eq_smul_span_int (S : set R) (I : ideal R) :
(↑(S • I) : set R) = (↑(S • (span ℤ (↑I : set R))) : set R) :=
begin
conv_lhs {erw ← span_eq I},
dsimp only [(•)],
erw [span_mul_span, span_mul_span],
apply set.subset.antisymm,
all_goals {
intros x hx,
apply span_induction hx,
{ intros, apply subset_span, assumption },
{ apply submodule.zero_mem (span _ _) },
{ intros, apply submodule.add_mem (span _ _), assumption' },
{ intros a si hsi,
apply span_induction hsi,
{ rintros _ ⟨s, hs, i, hi, rfl⟩,
apply subset_span,
refine ⟨s, hs, a * i, I.mul_mem_left hi, _⟩,
rw [← mul_assoc, mul_comm s a, mul_assoc, smul_def''],
refl },
{ rw smul_zero, apply submodule.zero_mem (span _ _) },
{ intros, rw smul_add, apply submodule.add_mem (span _ _), assumption' },
{ intros b si hsi,
rw [show a • b • si = b • a • si, by {simp}],
apply submodule.smul_mem (span _ _) b hsi } } }
end
section
variables {B : Type*} [comm_ring B] [algebra R B]
variables (S : subalgebra R B)
lemma span_mono' (X : set B) : (↑(span R X) : set B) ⊆ span S X :=
λ b hb, span_induction hb
(λ x hx, subset_span hx)
(span S X).zero_mem
(λ x y hx hy, (span S X).add_mem hx hy)
(λ r b hb, by { rw algebra.smul_def, exact (span S X).smul_mem (algebra_map S r) hb })
lemma span_span' (X : set B) : span S ↑(span R X) = span S X :=
le_antisymm (span_le.mpr $ span_mono' S X) (span_mono subset_span)
lemma span_span_int (S' : set B) [is_subring S'] (X : set B) : span S' ↑(span ℤ X) = span S' X :=
le_antisymm
begin
rw span_le,
intros x hx,
refine span_induction hx (λ x hx, subset_span hx) (span S' X).zero_mem
(λ x y hx hy, (span S' X).add_mem hx hy) _,
intros n b hb,
erw [smul_def'', ← gsmul_eq_mul],
apply is_add_subgroup.gsmul_mem hb,
end
(span_mono subset_span)
end
instance mul_action_algebra : mul_action A (submodule R A) :=
{ smul := λ a M, ({a} : set A) • M,
mul_smul := λ s t P, show ({s * t} : set A) • _ = _,
by { rw [is_mul_hom.map_mul (singleton : A → set A)], apply mul_smul },
one_smul := (submodule.semimodule_set R A).one_smul }
end submodule
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