Theorems · Definition · commutative algebra

IsFractionRing.semilinearEquivOfRingEquiv

{A : Type u_8} →
  {B : Type u_9} →
    (K : Type u_10) →
      (L : Type u_11) →
        [inst : CommRing A] →
          [inst_1 : CommRing B] →
            [inst_2 : CommRing K] →
              [inst_3 : CommRing L] →
                [inst_4 : Algebra A K] →
                  [IsFractionRing A K] → [inst_6 : Algebra B L] → [IsFractionRing B L] → (f : A ≃+* B) → K ≃ₛₗ[↑f] L

Given rings A, B and localization maps to their fraction rings f : A →+* K, g : B →+* L, an isomorphism h : A ≃+* B induces a semilinear equivalence fraction rings K ≃ₛₗ[f.toRingHom] L.

Defined in

Mathlib.RingTheory.Localization.FractionRing

Cited by

13 results in Mathlib

Foundations

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

Assumes

CommRingCommRingCommRingCommRingAlgebraIsFractionRingAlgebraIsFractionRing