Theorems · Definition · commutative algebra

IsLocalization.ringEquivOfRingEquiv

{R : Type u_1} →
  [inst : CommSemiring R] →
    {M : Submonoid R} →
      (S : Type u_2) →
        [inst_1 : CommSemiring S] →
          [inst_2 : Algebra R S] →
            {P : Type u_3} →
              [inst_3 : CommSemiring P] →
                [IsLocalization M S] →
                  {T : Submonoid P} →
                    (Q : Type u_4) →
                      [inst_5 : CommSemiring Q] →
                        [inst_6 : Algebra P Q] →
                          [IsLocalization T Q] → (h : R ≃+* P) → Submonoid.map h.toMonoidHom M = T → S ≃+* Q

If S, Q are localizations of R and P at submonoids M, T respectively, an isomorphism j : R ≃+* P such that j(M) = T induces an isomorphism of localizations S ≃+* Q.

Defined in

Mathlib.RingTheory.Localization.Defs

Cited by

15 results in Mathlib

Foundations

Depth 58 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringCommSemiringAlgebraCommSemiringIsLocalizationCommSemiringAlgebraIsLocalization