Theorems · Theorem · commutative algebra

LinearMap.lTensor_inj_iff_rTensor_inj

∀ {R : Type u_1} [inst : CommSemiring R] (M : Type u_7) {N : Type u_8} {P : Type u_9} [inst_1 : AddCommMonoid M]
  [inst_2 : AddCommMonoid N] [inst_3 : AddCommMonoid P] [inst_4 : Module R M] [inst_5 : Module R N]
  [inst_6 : Module R P] (f : N →ₗ[R] P),
  Function.Injective ⇑(LinearMap.lTensor M f) ↔ Function.Injective ⇑(LinearMap.rTensor M f)

Given a linear map f : N → P, f ⊗ M is injective if and only if M ⊗ f is injective.

Defined in

Mathlib.LinearAlgebra.TensorProduct.Map

Cited by

3 results in Mathlib

Foundations

Depth 64 from the axioms · uses propext, Quot.sound

Assumes

CommSemiringAddCommMonoidAddCommMonoidAddCommMonoidModuleModuleModule