Theorems · Definition · linear algebra

Submodule.giMapComap

{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} →
                          Function.Surjective ⇑f → GaloisInsertion (Submodule.map f) (Submodule.comap f)

map f and comap f form a GaloisInsertion when f is surjective.

Defined in

Mathlib.Algebra.Module.Submodule.Map

Cited by

11 results in Mathlib

Foundations

Depth 28 from the axioms · uses propext, Quot.sound

Assumes

SemiringSemiringAddCommMonoidAddCommMonoidModuleModuleRingHomSurjective