Theorems · Theorem · commutative algebra

LinearMap.ker_eq_bot_of_cancel

∀ {R : Type u_1} {R₂ : Type u_2} {M : Type u_5} {M₂ : Type u_6} [inst : Semiring R] [inst_1 : Semiring R₂]
  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] [inst_4 : Module R M] [inst_5 : Module R₂ M₂] {τ₁₂ : R →+* R₂}
  {f : M →ₛₗ[τ₁₂] M₂}, (∀ (u v : ↥f.ker →ₗ[R] M), f ∘ₛₗ u = f ∘ₛₗ v → u = v) → f.ker = ⊥

A monomorphism is injective.

Defined in

Mathlib.Algebra.Module.Submodule.Range

Cited by

1 results in Mathlib

Foundations

Depth 65 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringSemiringAddCommMonoidAddCommMonoidModuleModule