Theorems · Definition · linear algebra

LinearMap.submoduleMap

{R : Type u_1} →
  {R₂ : Type u_3} →
    {M : Type u_5} →
      {M₂ : Type u_7} →
        [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] →
                    {σ₁₂ : R →+* R₂} →
                      [inst_6 : RingHomSurjective σ₁₂] →
                        (f : M →ₛₗ[σ₁₂] M₂) → (p : Submodule R M) → ↥p →ₛₗ[σ₁₂] ↥(Submodule.map f p)

A linear map between two modules restricts to a linear map from any submodule p of the domain onto the image of that submodule. This is the linear version of AddMonoidHom.addSubmonoidMap and AddMonoidHom.addSubgroupMap. TODO: Consider making this an abbrev, dropping its API, and renaming to something like restrictSubmodule.

Defined in

Mathlib.Algebra.Module.Submodule.Map

Cited by

13 results in Mathlib

Foundations

Depth 29 from the axioms · uses propext, Quot.sound

Assumes

SemiringSemiringAddCommMonoidAddCommMonoidModuleModuleRingHomSurjective