Forcing linearity numbers of semicyclic modules over commutative Noetherian rings
type: Item
instance of: scholarly article
title: Forcing linearity numbers of semicyclic modules over commutative Noetherian rings
zbMATH Open document ID: 1087.13505
DOI: 10.1007/BF02941277
author: A. Moch Ryan
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 2004-05-18T00:00:00Z
review text: Given a module \(V\) over a commutative ring \(R\), a function \(f:V\rightarrow V\) is said to be ``homogeneous'' if \(f(rv) = rf(v)\) for all \(r\in R\) and all \(v\in V\) and \(M_R(V)\) denotes the set of all such functions. Clearly \(M_R(V)\) is contained in \(\text{End}_R(V)\), the set of module endomorphisms on \(V\). This paper looks at how close these two sets are in the case where \(R\) is Noetherian and \(V\) is semicyclic, meaning that \(V\) is decomposable as a direct sum of cyclic submodules. This closeness is measured by the ``forcing linearity number'' as defined by \textit{C. J. Maxson} and \textit{J. H. Meyer} [J. Algebra 223, 190--207 (2000; Zbl 0953.16034)], based on the minimum number of submodules of \(V\) on which there is a homogeneous \(f\) which is not linear. The author here extends some results for finitely generated modules obtained by \textit{C. J. Maxson} and \textit{A. B. Van der Merwe} [Rocky Mt. J. Math. 35, 929--939 (2005; Zbl 1098.13020)] to the non-finitely generated case, as well as other recent results by Maxson and Meyer.
Mathematics Subject Classification ID: 13B10
Mathematics Subject Classification ID: 13E05
Mathematics Subject Classification ID: 13C05
zbMATH DE Number: 2063944
zbMATH Keywords: homogeneous function
zbMATH Keywords: linearity of endomorphism
zbMATH Keywords: cyclic submodule
zbMATH Keywords: Noetherian ring
reviewed by: John Clark
MaRDI profile type: Publication
cites work: Q5563439
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full work available at URL: https://doi.org/10.1007/bf02941277
OpenAlex ID: W2004364086
description: scientific article; zbMATH DE number 2063944