Theorems · Theorem · commutative algebra

Module.Flat.lTensor_preserves_injective_linearMap

∀ {R : Type u} {M : Type v} {N : Type u_1} {P : Type u_2} [inst : CommSemiring R] [inst_1 : AddCommMonoid M]
  [inst_2 : Module R M] [inst_3 : AddCommMonoid N] [inst_4 : Module R N] [inst_5 : AddCommMonoid P]
  [inst_6 : Module R P] [Module.Flat R M] (f : N →ₗ[R] P),
  Function.Injective ⇑f → Function.Injective ⇑(LinearMap.lTensor M f)

If M is a flat module, then 𝟙 M ⊗ f is injective for all injective linear maps f.

Defined in

Mathlib.RingTheory.Flat.Basic

Cited by

31 results in Mathlib

Foundations

Depth 102 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringAddCommMonoidModuleAddCommMonoidModuleAddCommMonoidModuleModule.Flat