Quintasymptotic primes and four results of Schenzel
type: Item
instance of: scholarly article
title: Quintasymptotic primes and four results of Schenzel
zbMATH Open document ID: 0641.13005
DOI: 10.1016/0022-4049(87)90052-1
author: Stephen McAdam
published in: Journal of Pure and Applied Algebra
publication date: 1987-01-01T00:00:00Z
review text: Let I, resp. S, denote an ideal, resp. a multiplicatively closed set of a Noetherian ring R. By \(T_ 1, T_ 2, T_ 3\) denote the topologies defined by the filtrations \(\{I^ nR_ S\cap R\}\), \(\{I^ n\}\), \(\{(I^ n)_ a\}\), \(n\geq 1\). Here \(I_ a\) denotes the integral closure of I. In the reviewer's paper [Math. Nachr. 129, 123-148 (1986; Zbl 0606.13001)], among others, it is solved when \(T_ 1\) is equivalent resp. linearly equivalent to either of \(T_ 2\) or \(T_ 3\). In the paper under review the author gives a concise proof of these results. In fact, the arguments follow more or less the same line based on a variation of a theorem of Chevalley [\textit{O. Zariski} and \textit{P. Samuel}, ``Commutative algebra'', Vol. II (Princeton 1960; Zbl 0121.278); reprint 1976] and its application to Rees rings. In contrast to the reviewer's paper cited above the author uses consequently the notion of several types of asymptotic prime divisors in order to describe the main results. Furthermore he considers new relations among the various types of asymptotic prime divisors resp. differences between them by examples. Finally he continues with the study of so-called quintasymptotic prime divisors begun by the reviewer (loc. cit.) under a different notation.
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13J99
zbMATH DE Number: 4043969
zbMATH Keywords: adic topology
zbMATH Keywords: topologies defined by the filtrations
zbMATH Keywords: Rees rings
zbMATH Keywords: asymptotic prime divisors
reviewed by: Peter Schenzel
MaRDI profile type: Publication
full work available at URL: 0022-4049(87)90052-1
OpenAlex ID: W2023905020
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description: scientific article; zbMATH DE number 4043969