Baire's category theorem and prime avoidance in complete local rings

Sharp, Rodney Y.; Vámos, Peter

doi:10.1007/bf01237858

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Prime Ideals and Maximal Filters1

Introduction1

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1992

2022

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Cited by 37 publications

(30 citation statements)

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“…(ii) If R is a ring containing an uncountable field, then any countable set P of prime ideals has the property that every finitely generated ideal contained in a union of prime ideals in P is contained in one of these prime ideals [23]. Thus if L is a weakly saturated lattice on P ⊆ Spec(R) such that each element V ∈ L is countable, then L is saturated.…”

Section: Prime Ideals and Maximal Filtersmentioning

confidence: 96%

Prime ideals in ultraproducts of commutative rings

Olberding

Shapiro

2005

Journal of Algebra

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Section: Prime Ideals and Maximal Filtersmentioning

confidence: 96%

Prime ideals in ultraproducts of commutative rings

Olberding

Shapiro

2005

Journal of Algebra

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“…One of the fundamental cornerstones of commutative ring theory is the "prime avoidance" theorem, which states that, if p 1 ,...,p n are prime ideals of R and a is an ideal of R such that a ⊆ n i=1 p i , then a ⊆ p j for some 1 ≤ j ≤ n. In [4], the authors proved and used an extension of the following. Theorem 1.1.…”

Section: Introductionmentioning

confidence: 99%

A note on the countable union of prime submodules

Pournaki

Tousi

2001

International Journal of Mathematics and Mathematical Sciences

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Abstract. Let M be a finitely-generated module over a Noetherian ring R. Suppose a is an ideal of R and let N = aM and, M is complete with respect to the b-adic topology, {P i } i≥1 is a countable family of prime submodules of M, and x ∈ M, then x + N ⊆ i≥1 P i implies that x + N ⊆ P j for some j ≥ 1. This extends a theorem of Sharp and Vámos concerning prime ideals to prime submodules.

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“…Lemma 1.2 (countable prime avoidance [5,Lemma 3]; see also [13] Proof. Assume that the singular locus of R has dimension greater than one.…”

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confidence: 99%

Local rings of countable Cohen-Macaulay type

Huneke

Leuschke

2003

Proc. Amer. Math. Soc.

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Abstract. We prove (the excellent case of) Schreyer's conjecture that a local ring with countable CM type has at most a one-dimensional singular locus. Furthermore, we prove that the localization of a Cohen-Macaulay local ring of countable CM type is again of countable CM type. We verify Conjecture 0.1 more generally for all excellent CM local rings satisfying countable prime avoidance (Lemma 1.2). Some assumption of uncountability is Let (R,

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Baire's category theorem and prime avoidance in complete local rings

Cited by 37 publications

References 4 publications

Prime ideals in ultraproducts of commutative rings

Prime ideals in ultraproducts of commutative rings

A note on the countable union of prime submodules

Local rings of countable Cohen-Macaulay type

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