Asymptotic Prime Divisors

McAdam, Stephen

doi:10.1007/bfb0071575

Cited by 173 publications

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“…Suppose this is not the case. Then, by [13,Theorem 7.4] and [14,Lemma 3.15], it is easy to see that (IT : T q ) ⊆ q k , which is a contradiction. So q ∩ R ∈ Q * (Φ) and (i) follows.…”

Section: Proof (I) Let Q ∈ Q * (φT ) It Follows From Proposition 2mentioning

confidence: 99%

“…Also, one easily sees that [18,Lemma 2.3]). Therefore we may assume that R is local at p. Furthermore, by [14,Lemma 3.15], we may assume in addition in view of [13,Theorem 7.4] and Theorem 2.5 that R is complete. Now, since p ∈ Q * (Φ), we see that p ∩ S = ∅.…”

Section: Proof (I) Let Q ∈ Q * (φT ) It Follows From Proposition 2mentioning

confidence: 99%

“…Then, for such k, there is an ideal I k ∈ Φ with I k : R p ⊆ (p k ) a . Then by [13, Theorem 7.4(iii)] and [14,Lemma 3.15] we deduce that x ∈ I k R * : R * pR * . Because of Rad(I k R * : R * pR * + z) = pR * by [15, Lemma 3.1] we have pR * ∈ Ass R * R * /I k R * : R * pR * .…”

mentioning

confidence: 91%

“…Therefore, for each integer k ≥ 0 there is J ∈ Φ such that (JR * ) a : R * pR * ⊆ (p k R * ) a = (pR * ) k ; note that pR * is the unique maximal ideal of R * . Now, it is easy to see that [14,Lemma 3.15] provides a contradiction.…”

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Quintasymptotic primes, local cohomology and ideal topologies

2006

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Abstract. Let Φ be a system of ideals on a commutative Noetherian ring R, and let S be a multiplicatively closed subset of R. The first result shows that the topologies defined by {I a } I∈Φ and {S(I a )} I∈Φ are equivalent if and only if S is disjoint from the quintasymptotic primes of Φ. Also, by using the generalized Lichtenbaum-Hartshorne vanishing theorem we show that, if (R, m) is a d-dimensional local quasi-unmixed ring, then H d Φ (R), the dth local cohomology module of R with respect to Φ, vanishes if and only if there exists a multiplicatively closed subset S of R such that S ∩ m = ∅ and the S(Φ)-topology is finer than the Φ a -topology.1. Introduction. Throughout this paper, all rings considered will be commutative and Noetherian and will have non-zero identity elements. Such a ring will be denoted by R and a typical ideal of R will be denoted by I. Let (Λ, ≤) be a (non-empty) directed partially ordered set. A system of ideals of R over Λ is an inverse family Φ = {I α : α ∈ Λ} of ideals of R with the additional property that, for all α, γ ∈ Λ, there exists δ ∈ Λ such that I δ ⊆ I α I γ . Systems of ideals are a very useful generalization of the sets of powers of an ideal I in a ring R, and there are many important systems of ideals that are not powers. They have played an important role in many research papers, and there are numerous results concerning them in the literature (e.g., see [2], [3] and [7]).Let Φ denote a system of ideals (of R) and S a multiplicatively closed subset of R. For an ideal I of R, the S-component of I, denoted by S(I), is defined to be the union of (I : R s), where s varies in S. The integral closure of I in R is the ideal

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Section: Proof (I) Let Q ∈ Q * (φT ) It Follows From Proposition 2mentioning

confidence: 99%

Section: Proof (I) Let Q ∈ Q * (φT ) It Follows From Proposition 2mentioning

confidence: 99%

mentioning

confidence: 91%

mentioning

confidence: 99%

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Quintasymptotic primes, local cohomology and ideal topologies

2006

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“…The above-mentioned results of Brodmann and Ratliff have led to a large body of research: see, for example, McAdam's book [7]. Indeed, that research provides ideas for possible directions in which the theory of asymptotic behaviour of ideals relative to injective /4-modules might be pursued.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic behaviour of ideals relative to injective modules over commutative Noetherian rings II

Ansari-Toroghy¹,

Sharp²

1992

Proceedings of the Edinburgh Mathematical Society

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Let £ be an injective module over a commutative Noetherian ring A (with non-zero identity), and let a be an ideal of A. The submodule (0: £ a) of E has a secondary representation, and so we can form the finite set Att. A (0: E a) of its attached prime ideals. In [1, 3.1], we showed that the sequence of sets (Att / ,(0. E o")) Be M is ultimately constant; in [2], we introduced the integral closure a* (£) of o relative to E, and showed that (Att^(0: E (o")* (E) )) ne M is increasing and ultimately constant. In this paper, we prove that, if a contains an element r such that rE = E, then (Att / ,((0: £ a")/(0: £ (a°)* (£) ))) n6N is ultimately constant, and we obtain information about its ultimate constant value.1991 Mathematics subject classification: 13E10.

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Finiteness of Relative REES Rings and Asymptotic Prime Divisors

Schenzel

1986

Mathematische Nachrichten

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Asymptotic Prime Divisors

Cited by 173 publications

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Quintasymptotic primes, local cohomology and ideal topologies

Quintasymptotic primes, local cohomology and ideal topologies

Asymptotic behaviour of ideals relative to injective modules over commutative Noetherian rings II

Finiteness of Relative REES Rings and Asymptotic Prime Divisors

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