Theorems · Theorem · commutative algebra

IsLocalization.AtPrime.equivQuotientMapOfIsMaximal_apply_mk

∀ {R : Type u_1} {S : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] (p : Ideal R)
  [inst_3 : p.IsPrime] (Sₚ : Type u_4) [inst_4 : CommRing Sₚ] [inst_5 : Algebra S Sₚ]
  [inst_6 : IsLocalization (Algebra.algebraMapSubmonoid S p.primeCompl) Sₚ] (P : Ideal S) [hPp : P.LiesOver p]
  [inst_7 : P.IsMaximal] (x : S),
  (IsLocalization.AtPrime.equivQuotientMapOfIsMaximal p Sₚ P) ((Ideal.Quotient.mk P) x) =
    (Ideal.Quotient.mk (Ideal.map (algebraMap S Sₚ) P)) ((algebraMap S Sₚ) x)

Defined in: Mathlib.RingTheory.Localization.AtPrime.Extension
Cited by: 1 results in Mathlib
Foundations: Depth 95 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommRing CommRing Algebra Ideal.IsPrime CommRing Algebra IsLocalization Ideal.LiesOver Ideal.IsMaximal

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Cites17

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

DFunLike.coestatement · cited by 62,936
CommRingstatement and proof · cited by 17,173
Algebrastatement and proof · cited by 11,388
RingHomstatement · cited by 10,189
Idealstatement and proof · cited by 4,748
Algebra.algebraMapstatement · cited by 4,706
HasQuotient.Quotientstatement · cited by 2,301
RingEquivstatement · cited by 1,147
Ideal.IsPrimestatement and proof · cited by 827
Ideal.mapstatement · cited by 692
IsLocalizationstatement and proof · cited by 636
Ideal.Quotient.mkstatement · cited by 610

Cited by1

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IsLocalization.AtPrime.equivQuotientMapOfIsMaximal_symm_apply_mkproof · cited by 1