Rank-sets of indecomposable modules over one-dimensional rings
type: Item
instance of: scholarly article
title: Rank-sets of indecomposable modules over one-dimensional rings
zbMATH Open document ID: 1151.13010
author: Melissa R. Luckas
published in: Journal of Pure and Applied Algebra
publication date: 2008-10-21T00:00:00Z
review text: A ring \(R\) is a \textit{ring-order}, if it is a reduced, one-dimensional, commutative noetherian ring with integral closure a finitely generated \(R-\)module. Let \(M\) be a finitely generated, torsion-free \(R-\)module. If \(P_1,\ldots,P_s\) are the minimal prime ideals of \(R\), define the \textit{rank} of \(M\) to be \(rank(M)=(r_1,\ldots,r_s),\) where \(r_i:=\dim_{R_{P_i}}(M_{P_i}), 1\leq i\leq s\). The set \(\text{Ranks}(M)=\{r_1,\ldots,r_s\}\) is called the rank-set of \(M\). \(M\) has \textit{constant rank} \(N\) if \(\text{rank}(M)=(N,\ldots,N)\). \(R\) has \textit{bounded representation type} if there exists an \(N\) such that for each indecomposable finitely generated torsion-free \(R-\)module \(M\), \(rank(M)\leq (N,\ldots,N)\). The article fits into the problem of describing the isomorphism classes of indecomposable modules over one-dimensional rings, by investigating what subsets of the integers are realizable as the rank-sets of indecomposable modules (Question 1.3). By \textit{M. Arnavut, M. Luckas} and \textit{S. Wiegand} [J. Pure Appl. Algebra 208, No. 2, 739--760 (2007; Zbl 1111.13007)] and [\textit{N. Baeth, M. Luckas}, Bounds for indecomposable torsion-free modules, J. Pure Appl. Algebra (submitted)], the rank-set of every indecomposable module is either a set or a proper subset of a set listed in Table 1.5. For example, from this table, there are no indecomposable modules of constant rank 5. The Main Theorem 1.6. guarantees that for every set in Table 1.5. there exist a semilocal ring-order \(R\) of bounded representation type and an indecomposable finitely generated torsion-free \(R-\)module with rank-set exactly that set. The key of the Main Theorem is Proposition 3.1 which enables to construct indecomposable modules of rank \((a_1,\ldots,a_s,t), t\leq 2a_{\ell}\) for some \(1\leq\ell\leq s\), from indecomposable modules of rank \((a_1,\ldots,a_s)\). The base for this construction are three local ring-orders of bounded representation type de…
Mathematics Subject Classification ID: 13C14
Mathematics Subject Classification ID: 13H10
zbMATH DE Number: 5354972
zbMATH Keywords: ring-order
zbMATH Keywords: indecomposable modules
zbMATH Keywords: rank-set
zbMATH Keywords: minimal prime ideals
zbMATH Keywords: semilocal rings
reviewed by: Stefan O. Tohǎneanu
MaRDI profile type: Publication
full work available at URL: j.jpaa.2008.05.012
OpenAlex ID: W2050384185
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DOI: 10.1016/J.JPAA.2008.05.012
description: scientific article; zbMATH DE number 5354972