Asymptotic behaviour of integral closures of ideals relative to Artinian modules
type: Item
instance of: scholarly article
title: Asymptotic behaviour of integral closures of ideals relative to Artinian modules
zbMATH Open document ID: 0770.13010
DOI: 10.1016/0021-8693(92)90155-F
published in: Journal of Algebra
publication date: 1993-03-28T00:00:00Z
review text: Let \(I\) be an ideal of a commutative ring \(R\) and let \(A\) be an Artinian \(R\)-module. Denote by \(I^*\) the integral closure of \(I\) relative to \(A\) as introduced by \textit{R. Y. Sharp} and \textit{A.-J. Taherizadeh} [J. Lond. Math. Soc., II. Ser. 37, No. 2, 203-218 (1988; Zbl 0656.13001)]. The structure of \(A\) over \(R\) is closely related to its structure over a certain complete semilocal Noetherian ring \(S\) [compare \textit{R.Y. Sharp}, Math. Proc. Camb. Philos. Soc. 111, No. 1, 25-33 (1992; Zbl 0758.13007)]. By setting up a Matlis duality over \(S\) and using a result of Ratliff on asymptotic primes, it is shown that, for an ideal \(J\) of \(S\), the sequence \(\text{Ass}((J^ n)^*)\) is increasing and ultimately constant. In \(R\), it follows that \(\bigcup_ n\text{Ass}((I^ n)^*)\) is finite.
Mathematics Subject Classification ID: 13E10
Mathematics Subject Classification ID: 13B22
Mathematics Subject Classification ID: 13A30
zbMATH DE Number: 130469
zbMATH Keywords: asymptotic prime ideals
zbMATH Keywords: Artinian module
zbMATH Keywords: integral closure
zbMATH Keywords: Matlis duality
author: R. Y. Sharp
author: Yücel Tiraş
reviewed by: C. P. L. Rhodes
MaRDI profile type: Publication
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full work available at URL: 0021-8693(92)90155-f
OpenAlex ID: W2042513711
description: scientific article; zbMATH DE number 130469