The Laskerian condition for nonfinitely generated modules
type: Item
instance of: scholarly article
title: The Laskerian condition for nonfinitely generated modules
zbMATH Open document ID: 0784.13009
DOI: 10.1007/BF02454378
author: Helmut Zoeschinger
published in: Periodica Mathematica Hungarica
publication date: 1993-02-08T00:00:00Z
review text: Let \(R\) be a commutative ring and let \(M\) be an \(R\)-module. \(M\) is said to have property \((L)\) if every submodule of \(M\) is an intersection of finitely many submodules \(V_ i\) of \(R\) such that each \(M/V_ i\) is primary. Finitely generated modules with this property are called Laskerian in the literature. The main result in the first section of this paper is that if \(R\) is noetherian, then \(M\) has property \((L)\) if and only if it is a sum \(A+B\) of submodules, with \(A\) finitely generated, and \(R/\text{Ann}_ R(B)\) artinian. -- The second section gives various characterizations of modules that have a primary decomposition, again in the case of a noetherian base ring. -- The final section of the paper provides examples, with particular attention to the case where \(R\) is of dimension one. Modules that have both primary and coprimary decompositions are also characterized.
reviewed by: William H. Gustafson
Mathematics Subject Classification ID: 13E99
Mathematics Subject Classification ID: 13C05
Mathematics Subject Classification ID: 13C13
Mathematics Subject Classification ID: 13E05
zbMATH DE Number: 118333
zbMATH Keywords: Laskerian modules
zbMATH Keywords: primary decomposition
MaRDI profile type: Publication
cites work: Q5531991
cites work: The Laskerian property in commutative rings
cites work: Coprimary Decomposition of Artinian Modules
cites work: Modules satisfying ACC on a certain type of colons
cites work: Q3215335
cites work: Secondary representations for injective modules over commutative Noetherian rings
full work available at URL: https://doi.org/10.1007/bf02454378
OpenAlex ID: W2066050703
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description: scientific article; zbMATH DE number 118333