Theorems · Theorem · commutative algebra
PrimeSpectrum.iInf_localization_eq_bot
∀ (R : Type u_4) [inst : CommRing R] [inst_1 : IsDomain R] (K : Type u_5) [inst_2 : Field K] [inst_3 : Algebra R K] [inst_4 : IsFractionRing R K], ⨅ v, Localization.subalgebra.ofField K v.asIdeal.primeCompl ⋯ = ⊥
An integral domain is equal to the intersection of its localizations at all its prime ideals viewed as subalgebras of its field of fractions.
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- Foundations
- Depth 80 from the axioms · uses propext, Classical.choice, Quot.sound
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- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- Fieldstatement and proof · cited by 7,404
- Idealproof · cited by 4,748
- Bot.botstatement · cited by 4,720
- LE.le.transproof · cited by 3,151
- IsDomainstatement and proof · cited by 2,196
- iInfstatement · cited by 1,690
- Subalgebrastatement · cited by 1,353
- IsFractionRingstatement and proof · cited by 738
- PrimeSpectrumstatement and proof · cited by 625
- Eq.leproof · cited by 605
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