Theorems · Definition · category theory

lequivProdOfRightSplitExact

{R : Type u} →
  {A : Type uA} →
    {M : Type uM} →
      {B : Type uB} →
        [inst : Ring R] →
          [inst_1 : AddCommGroup A] →
            [inst_2 : AddCommGroup B] →
              [inst_3 : AddCommGroup M] →
                [inst_4 : Module R A] →
                  [inst_5 : Module R B] →
                    [inst_6 : Module R M] →
                      {j : A →ₗ[R] M} →
                        {g : M →ₗ[R] B} →
                          {f : B →ₗ[R] M} →
                            Function.Injective ⇑j → j.range = g.ker → g ∘ₗ f = LinearMap.id → (A × B) ≃ₗ[R] M

The isomorphism A × B ≃ₗ[R] M coming from a right split exact sequence 0 ⟶ A ⟶ M ⟶ B ⟶ 0 of modules.

Defined in

Mathlib.Algebra.Category.ModuleCat.Biproducts

Cited by

2 results in Mathlib

Foundations

Depth 106 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingAddCommGroupAddCommGroupAddCommGroupModuleModuleModule