Theorems · Definition · linear algebra

Submodule.equivMapOfInjective

{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} →
                      {σ₂₁ : R₂ →+* R} →
                        [inst_6 : RingHomInvPair σ₁₂ σ₂₁] →
                          [inst_7 : RingHomInvPair σ₂₁ σ₁₂] →
                            (f : M →ₛₗ[σ₁₂] M₂) →
                              Function.Injective ⇑f → (p : Submodule R M) → ↥p ≃ₛₗ[σ₁₂] ↥(Submodule.map f p)

The pushforward of a submodule by an injective linear map is linearly equivalent to the original submodule. See also LinearEquiv.submoduleMap for a computable version when f has an explicit inverse.

Defined in

Mathlib.Algebra.Module.Submodule.Map

Cited by

17 results in Mathlib

Foundations

Depth 26 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringSemiringAddCommMonoidAddCommMonoidModuleModuleRingHomInvPairRingHomInvPair