Theorems · Theorem · commutative algebra
IsIntegrallyClosed.of_localization_maximal
∀ {R : Type u_1} [inst : CommRing R] [IsDomain R],
(∀ (p : Ideal R), p ≠ ⊥ → ∀ [inst_2 : p.IsMaximal], IsIntegrallyClosed (Localization.AtPrime p)) →
IsIntegrallyClosed RAn integral domain R is integral closed if Rₘ is integral closed
for any maximal ideal m of R.
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 122 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites27
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommRingstatement and proof · cited by 17,173
- Idealstatement and proof · cited by 4,748
- Bot.botstatement and proof · cited by 4,720
- Set.rangeproof · cited by 4,705
- IsDomainstatement and proof · cited by 2,196
- iInfproof · cited by 1,690
- Subalgebraproof · cited by 1,353
- PrimeSpectrumproof · cited by 625
- Ideal.primeComplstatement and proof · cited by 462
- Ideal.IsMaximalstatement and proof · cited by 452
- PrimeSpectrum.asIdealproof · cited by 333
- Localization.AtPrimestatement and proof · cited by 299
Cited by2
Results whose statement or proof uses this declaration.
- IsIntegrallyClosed.of_isLocalization_maximalproof · cited by 1
- isIntegrallyClosed_ofLocalizationMaximalproof · cited by 0