PrimeSpectrum.iInf_localization_eq_bot
theorem PrimeSpectrum.iInf_localization_eq_bot :
∀ (R : Type u) [inst : CommRing R] [inst_1 : IsDomain R] (K : Type v) [inst_2 : Field K] [inst_3 : Algebra R K]
[inst_4 : IsFractionRing R K], ⨅ v, Localization.subalgebra.ofField K v.asIdeal.primeCompl ⋯ = ⊥
This theorem states that for any integral domain `R` and its field of fractions `K`, the intersection of all localizations of `R` at its prime ideals (viewed as subalgebras of `K`) is equal to the zero ideal. In other words, the meeting point of all these localizations is the trivial subalgebra of `R`, which contains only the zero element. This result underlines the importance of localizations at prime ideals in understanding the structure of an integral domain and its relationship with its field of fractions.
More concisely: The intersection of all localizations of an integral domain R at its prime ideals is the zero ideal in the field of fractions K.
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MaximalSpectrum.iInf_localization_eq_bot
theorem MaximalSpectrum.iInf_localization_eq_bot :
∀ (R : Type u) [inst : CommRing R] [inst_1 : IsDomain R] (K : Type v) [inst_2 : Field K] [inst_3 : Algebra R K]
[inst_4 : IsFractionRing R K], ⨅ v, Localization.subalgebra.ofField K v.asIdeal.primeCompl ⋯ = ⊥
This theorem states that for any integral domain `R` with the structure of a commutative ring and a field of fractions `K` (also denoted as the quotient field or field of fractions of `R`), the intersection of all localizations of `R` at its maximal ideals (viewed as subalgebras of its field of fractions `K`) is equal to the trivial subalgebra.
In mathematical terms, the theorem asserts that the intersection of all localizations of an integral domain at its maximal ideals, when we consider these localizations as subalgebras of the field of fractions, gives us the trivial algebra (represented by the bottom symbol ⊥, denoting the zero ideal or the trivial subalgebra in this context).
This theorem is a part of algebraic geometry and commutative algebra, particularly in studying the properties of rings and their spectrums.
More concisely: The intersection of all localizations of an integral domain at its maximal ideals, considered as subalgebras of its field of fractions, equals the trivial subalgebra.
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