Theorems · Definition · ring theory

AlgEquiv.ofAlgHom

{R : Type uR} →
  {A₁ : Type uA₁} →
    {A₂ : Type uA₂} →
      [inst : CommSemiring R] →
        [inst_1 : Semiring A₁] →
          [inst_2 : Semiring A₂] →
            [inst_3 : Algebra R A₁] →
              [inst_4 : Algebra R A₂] →
                (f : A₁ →ₐ[R] A₂) →
                  (g : A₂ →ₐ[R] A₁) → f.comp g = AlgHom.id R A₂ → g.comp f = AlgHom.id R A₁ → A₁ ≃ₐ[R] A₂

If an algebra morphism has an inverse, it is an algebra isomorphism.

Defined in

Mathlib.Algebra.Algebra.Equiv

Cited by

13 results in Mathlib

Foundations

Depth 25 from the axioms · uses propext, Quot.sound

Assumes

CommSemiringSemiringSemiringAlgebraAlgebra