Theorems · Definition · commutative algebra

Module.Invertible.linearEquivOfLeftInverse

{R : Type u} →
  {M : Type v} →
    {N : Type u_1} →
      [inst : CommSemiring R] →
        [inst_1 : AddCommMonoid M] →
          [inst_2 : AddCommMonoid N] →
            [inst_3 : Module R M] →
              [inst_4 : Module R N] →
                [Module.Invertible R M] →
                  [Module.Invertible R N] → {f : M →ₗ[R] N} → {g : N →ₗ[R] M} → Function.LeftInverse ⇑f ⇑g → M ≃ₗ[R] N

If f : M →ₗ[R] N and g : N →ₗ[R] M where M and N are invertible R-modules, and f is a left inverse of g, then in fact f is also the right inverse of g, and we promote this to an R-module isomorphism.

Defined in

Mathlib.RingTheory.PicardGroup

Cited by

2 results in Mathlib

Foundations

Depth 115 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringAddCommMonoidAddCommMonoidModuleModuleModule.InvertibleModule.Invertible