Finitely generated mixed modules of Warfield type
type: Item
instance of: scholarly article
title: Finitely generated mixed modules of Warfield type
zbMATH Open document ID: 1476.13031
DOI: 10.4171/RSMUP/71
author: P. Zanardo
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 2021-07-02T00:00:00Z
review text: Let \(R\) be a local one-dimensional domain which is not a valuation domain. The paper deals with a class of finitely generated mixed modules over \(R\) whose construction is similar to that used by \textit{R. B. Warfield jun.} [Proc. Am. Math. Soc. 25, 167--172 (1970; Zbl 0204.05902)] and called therefore of Warfield type. Theorem 2.1 shows that these \(R\)-modules have local endomorphism rings, hence they are indecomposable. The torsion part \(t(M)\) of a mixed module of Warfield type \(M\) is also examined: torsion elements of \(M\) are characterized in Theorem 3.1, then the author focuses on the natural property \(t(M)\subset \mathfrak{M}M\), where \(\mathfrak{M}\) is the maximal ideal of \(R\). Interestingly, if \(M\) is \(n\)-generated and defined by the elements \(a\) and \(b\) of \(R\), if either \(a/b\) or \(b/a\) is an integral element over \(R\) of degree \(\leq n-1\), then \(t(M)\) is not contained in \(\mathfrak{M}M\) (Theorem 3.3). In the last section of the paper it is shown that \(M/t(M)\) is isomorphic to an ideal of \(R\) and the minimum number of its generators is determined.
reviewed by: Laura Cossu
Mathematics Subject Classification ID: 13G05
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13A18
zbMATH DE Number: 7367423
zbMATH Keywords: local one-dimensional domains
zbMATH Keywords: finitely generated mixed modules
zbMATH Keywords: torsion elements
zbMATH Keywords: minimal sets of generators
MaRDI profile type: Publication
cites work: Q4056148
cites work: Q3994901
cites work: Almost perfect domains
cites work: Q3699812
cites work: Q4513600
cites work: Local domains with topologically T-nilpotent radical
cites work: Decomposability of Finitely Presented Modules
cites work: Direct-sum decompositions over local rings
cites work: ALMOST PERFECT LOCAL DOMAINS AND THEIR DOMINATING ARCHIMEDEAN VALUATION DOMAINS
full work available at URL: https://doi.org/10.4171/rsmup/71
OpenAlex ID: W3113259855
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description: scientific article; zbMATH DE number 7367423