Theorems · Definition · commutative algebra

AlgEquiv.extendScalarsOfSurjective

{R : Type u} →
  {S : Type v} →
    {A : Type w} →
      {B : Type u₁} →
        [inst : CommSemiring R] →
          [inst_1 : CommSemiring S] →
            [inst_2 : Semiring A] →
              [inst_3 : Semiring B] →
                [inst_4 : Algebra R S] →
                  [inst_5 : Algebra S A] →
                    [inst_6 : Algebra S B] →
                      [inst_7 : Algebra R A] →
                        [inst_8 : Algebra R B] →
                          [IsScalarTower R S A] →
                            [IsScalarTower R S B] → Function.Surjective ⇑(algebraMap R S) → (A ≃ₐ[R] B) ≃ A ≃ₐ[S] B

Any f : A ≃ₐ[R] B is also an S-algebra isomorphism if the R-algebra structure on A and B factors via a surjective ring homomorphism R →+* S.

Defined in

Mathlib.Algebra.Algebra.Tower

Cited by

5 results in Mathlib

Foundations

Depth 27 from the axioms · uses propext, Quot.sound

Assumes

CommSemiringCommSemiringSemiringSemiringAlgebraAlgebraAlgebraAlgebraAlgebraIsScalarTowerIsScalarTower