A Compactness Property for Prime Ideals in Noetherian Rings

A semigroup prime of a commutative ring R is a prime ideal of the semigroup (R, Â·). One of the purposes of this paper is to study, from a topological point of view, the space S(R) of prime semigroups of R. We show that, under a natural topology introduced by B. Olberding in 2010, S(R) is a spectral space (after Hochster), spectral extension of Spec(R) , and that the assignment Râ\u86¦ S(R) induces a contravariant functor. We then relateâ\u80\u94in the case R is an integral domainâ\u80\u94the topology on S(R) with the Zariski topology on the set of overrings of R. Furthermore, we investigate the relationship between S(R) and the space X(R) consisting of all nonempty inverse-closed subspaces of Spec(R) , which has been introduced and studied in Finocchiaro et al. (submitted). In this context, we show that S(R) is a spectral retract of X(R) and we characterize when S(R) is canonically homeomorphic to X(R) , both in general and when Spec(R) is a Noetherian space. In particular, we obtain that, when R is a BÃ©zout domain, S(R) is canonically homeomorphic both to X(R) and to the space Overr(R) of the overrings of R (endowed with the Zariski topology). Finally, we compare the space X(R) with the space S(R(T)) of semigroup primes of the Nagata ring R(T), providing a canonical spectral embedding X(R) â\u86ª S(R(T)) which makes X(R) a spectral retract of S(R(T))

show abstract

“…Remark 2.15. Rings verifying property (iii) of the previous corollary has been called compactly packed in [53].…”

Section: S(z) and S(z[x])mentioning

confidence: 99%

Topological properties of semigroup primes of a commutative ring

2017

View full text Add to dashboard Cite

show abstract

“…Infinite prime avoidance has been periodically investigated over the years, see e.g. [1], [8], [10], [12], and [13]. Dually, infinite prime absorbance has been studied in [9, §V] and [14, §4].…”

Section: Introductionmentioning

confidence: 99%

Avoidance and Absorbance

Tarizadeh

2020

Preprint

View full text Add to dashboard Cite

We study the two dual notions of prime avoidance and prime absorbance. We generalize the classical prime avoidance lemma to radical ideals. A number of new criteria are provided for an abstract ring to be C.P. (every set of primes satisfies avoidance) or P.Z. (every set of primes satisfies absorbance). Special consideration is given to the interaction with chain conditions and Noetherian-like properties. It is shown that a ring is both C.P. and P.Z. iff it has finite spectrum.

show abstract

“…Such rings were introduced under the name of compactly-packed (C.P.) rings in [4], and have been fairly well-studied, e.g. in [6], [3].…”

mentioning

confidence: 99%

Infinite prime avoidance

Chen

2017

Preprint

View full text Add to dashboard Cite

We investigate prime avoidance for an arbitrary set of prime ideals in a commutative ring. Various necessary and/or sufficient conditions for prime avoidance are given, which yield natural classes of infinite sets of primes that satisfy prime avoidance. Examples and counterexamples are given throughout to illustrate the phenomena that can occur. As an application, we show how to use prime avoidance to construct counterexamples among rings essentially of finite type.The classical prime avoidance lemma is one of the most ubiquitous results in commutative algebra. Prime avoidance, along with finiteness of associated primes, is one of the basic building blocks of the theory of Noetherian rings. For example, the two results can be jointly used to choose generic nonzerodivisors (such as in the converse of Krull's Altitude Theorem), or to select a single annihilator for an ideal consisting of zerodivisors.As a fundamental technical result, the prime avoidance lemma has found various extensions in the literature (cf. [2], [5]). Moreover, special cases of infinite prime avoidance have in the past been used to great effect, perhaps most famously as a crucial step in Nagata's example of an infinite-dimensional Noetherian ring. This indicates the potential utility of understanding and applying infinite prime avoidance methodically. The goal of this note is to make initial steps in this direction. To this end, we first make a definition. For a commutative ring R with 1 = 0, Spec R denotes the set (for now) of prime ideals of R.Definition. Let R be a ring, Λ ⊆ Spec R. We say that Λ satisfies prime avoidance if I ⊆ p∈Λ p =⇒ I ⊆ p for some p ∈ Λ, for any R-ideal I.Note that in the definition of prime avoidance, it is enough to check the condition for prime ideals I, since ideals which are maximal with respect to being contained in p∈Λ p (i.e. not meeting the multiplicative set R \

show abstract

A Compactness Property for Prime Ideals in Noetherian Rings

Cited by 5 publications

References 0 publications

Topological properties of semigroup primes of a commutative ring

Topological properties of semigroup primes of a commutative ring

Avoidance and Absorbance

Infinite prime avoidance

Contact Info

Product

Resources

About