isDedekindDomain — Mathlib

English

Localization at a prime ideal preserves Dedekind property: the localized ring is Dedekind.

Русский

Локализация поprime сохраняет свойство Дедекенд: локализованное кольцо Дедекендово.

LaTeX

$$$\forall \ A, P, [P.IsPrime] \Rightarrow IsDedekindDomain (Localization.AtPrime P)$$

Lean4

/-- The localization of a Dedekind domain at every nonzero prime ideal is a Dedekind domain. -/
theorem isDedekindDomain [IsDedekindDomain A] (P : Ideal A) [P.IsPrime] (Aₘ : Type*) [CommRing Aₘ] [IsDomain Aₘ]
    [Algebra A Aₘ] [IsLocalization.AtPrime Aₘ P] : IsDedekindDomain Aₘ :=
  IsLocalization.isDedekindDomain A P.primeCompl_le_nonZeroDivisors Aₘ

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