Index of reducibility of parameter ideals in a local ring
type: Item
instance of: scholarly article
title: Index of reducibility of parameter ideals in a local ring
zbMATH Open document ID: 0538.13003
DOI: 10.1016/0021-8693(84)90160-1
author: Shiro Goto
author: Naoyoshi Suzuki
published in: Journal of Algebra
publication date: 1984-01-01T00:00:00Z
review text: Let (A,m) be a local ring. If a is an m-primary ideal of A, then the number N(a) (called the index of reducibility of a) of irreducible ideals that appear in an irredundant decomposition of a is S(A/a) where S(M) denotes the dimension of the socle of an A-module M considered as an A/m- vector space. If a is a parameter ideal, certain properties of N(a) played an important role in the early development of Gorenstein and Cohen-Macaulay rings. For example, if A is Cohen-Macaulay, N(a) is an invariant of the ring. The purpose of this paper is to study N(a) for A not necessarily Cohen-Macaulay. For a finitely generated A-module M, define \(r(M)=\sup \{S(M/qM)| q\quad is\quad a\quad parameter\quad ideal\quad of\quad M\}\). Upper and lower bounds for r(M) involving local cohomologies for M are given (which become equalities if M is a quasi- Buchsbaum module). It is shown that if dim \(A\leq 3\), then \(r(A)<\infty\), but for any integer \(d>3\) there is a local ring A with \(\dim A=3\) and \(r(A)=\infty\). Rings with constant index of reducibility of parameter ideals are also studied.
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13E05
Mathematics Subject Classification ID: 13H10
zbMATH DE Number: 3855253
zbMATH Keywords: Noetherian local ring
zbMATH Keywords: index of reducibility of ideal
zbMATH Keywords: irredundant decomposition
zbMATH Keywords: quasi-Buchsbaum module
reviewed by: D. D. Anderson
MaRDI profile type: Publication
full work available at URL: 0021-8693(84)90160-1
OpenAlex ID: W2086904883
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description: scientific article; zbMATH DE number 3855253