Theorems · Definition · commutative algebra

IsFractionRing.fieldEquivOfAlgEquiv

{A : Type u_8} →
  {B : Type u_9} →
    {C : Type u_10} →
      [inst : CommRing A] →
        [inst_1 : CommRing B] →
          [inst_2 : CommRing C] →
            [inst_3 : Algebra A B] →
              [inst_4 : Algebra A C] →
                (FA : Type u_12) →
                  (FB : Type u_13) →
                    (FC : Type u_14) →
                      [inst_5 : Field FA] →
                        [inst_6 : Field FB] →
                          [inst_7 : Field FC] →
                            [inst_8 : Algebra A FA] →
                              [inst_9 : Algebra B FB] →
                                [inst_10 : Algebra C FC] →
                                  [IsFractionRing A FA] →
                                    [IsFractionRing B FB] →
                                      [IsFractionRing C FC] →
                                        [inst_14 : Algebra A FB] →
                                          [IsScalarTower A B FB] →
                                            [inst_16 : Algebra A FC] →
                                              [IsScalarTower A C FC] →
                                                [inst_18 : Algebra FA FB] →
                                                  [IsScalarTower A FA FB] →
                                                    [inst_20 : Algebra FA FC] →
                                                      [IsScalarTower A FA FC] → (B ≃ₐ[A] C) → FB ≃ₐ[FA] FC

An algebra isomorphism of rings induces an algebra isomorphism of fraction fields.

Defined in

Mathlib.RingTheory.Localization.FractionRing

Cited by

6 results in Mathlib

Foundations

Depth 62 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingCommRingAlgebraAlgebraFieldFieldFieldAlgebraAlgebraAlgebraIsFractionRingIsFractionRingIsFractionRingAlgebraIsScalarTowerAlgebraIsScalarTowerAlgebraIsScalarTowerAlgebraIsScalarTower