Theorems · Definition · linear algebra

Submodule.mapQLinear

{R : Type u_1} →
  {M : Type u_2} →
    {M₂ : Type u_3} →
      [inst : CommRing R] →
        [inst_1 : AddCommGroup M] →
          [inst_2 : Module R M] →
            [inst_3 : AddCommGroup M₂] →
              [inst_4 : Module R M₂] →
                (p : Submodule R M) → (q : Submodule R M₂) → ↥(p.compatibleMaps q) →ₗ[R] M ⧸ p →ₗ[R] M₂ ⧸ q

Given modules M, M₂ over a commutative ring, together with submodules p ⊆ M, q ⊆ M₂, the natural map $\{f ∈ Hom(M, M₂) | f(p) ⊆ q \} \to Hom(M/p, M₂/q)$ is linear.

Defined in

Mathlib.LinearAlgebra.Quotient.Basic

Cited by

0 results in Mathlib

Foundations

Depth 89 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleAddCommGroupModule