Theorems · Theorem · commutative algebra

IsLocalizedModule.map_injective_iff_localizedModuleMap_injective

∀ {R : Type u_1} {M : Type u_2} {N : Type u_3} {M' : Type u_4} {N' : Type u_5} [inst : CommSemiring R] {S : Submonoid R}
  [inst_1 : AddCommMonoid M] [inst_2 : Module R M] [inst_3 : AddCommMonoid N] [inst_4 : Module R N]
  [inst_5 : AddCommMonoid M'] [inst_6 : Module R M'] [inst_7 : AddCommMonoid N'] [inst_8 : Module R N']
  (g₁ : M →ₗ[R] M') (g₂ : N →ₗ[R] N') [inst_9 : IsLocalizedModule S g₁] [inst_10 : IsLocalizedModule S g₂]
  {l : M →ₗ[R] N},
  Function.Injective ⇑((IsLocalizedModule.map S g₁ g₂) l) ↔ Function.Injective ⇑((LocalizedModule.map S) l)

Defined in

Mathlib.RingTheory.Localization.Module

Cited by

0 results in Mathlib

Foundations

Depth 67 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringAddCommMonoidModuleAddCommMonoidModuleAddCommMonoidModuleAddCommMonoidModuleIsLocalizedModuleIsLocalizedModule