Applications of linear functionals to Kronecker modules. II
type: Item
instance of: scholarly article
title: Applications of linear functionals to Kronecker modules. II
zbMATH Open document ID: 0591.16012
DOI: 10.1016/0024-3795(86)90222-3
author: Frank Okoh
published in: Linear Algebra and its Applications
publication date: 1986-01-01T00:00:00Z
review text: This paper continues the investigation of part I [cf. the preceding review] - a class of modules \(E_ r\), \(r\geq 2\). In this paper a class of modules \(E_ r\), \(r\geq 2\) is constructed from submodules of the module of rational functions K(X), over the polynomial ring K[X], given by a height function h; sequences of linear functionals \(L=(\ell_ 2,...,\ell_ r)\) on K(X); and a sequence of positive integers \((m_ 2,...,m_ r)\). To each subset L' of \(\{\ell_ 2,...,\ell_ r\}\) is attached a field extension \(K_{L'}\) of the prime subfield of K. Let \(M_{L_ 1}\) and \(M_{L_ 2}\) be modules constructed from the sequences of linear functionals \(L_ 1\) and \(L_ 2\). The main results are as follows: If \(M_{L_ 1}\) is isomorphic to \(M_{L_ 2}\), then \(K_{L_ 1}\) and \(K_{L_ 2}\) are contained in finitely generated field extensions of each other. Let \(\bar L_ i=L\setminus \ell_ i\), \(i=2,...,r\). If \(M_ L\) is not purely simple, then for some i in \(\{\) 2,...,r\(\}\), \(K_{\ell_ i}\) is contained in a finitely generated extension of \(K_{\bar L_ i}\). These results are then used to obtain the following theorems: assuming that \(1<\sup p h<\infty\), then there are exactly \(\aleph_ 0\) isomorphism classes of completely decomposable modules in \(E_ r\); there are at least c (the cardinality of the continuum) isomorphism classes of (i) indecomposable but not purely simple modules in \(E_ r\), (ii) purely simple modules in \(E_ r\).
Mathematics Subject Classification ID: 16D80
Mathematics Subject Classification ID: 16Gxx
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 16D70
zbMATH DE Number: 3948525
zbMATH Keywords: module of rational functions
zbMATH Keywords: height function
zbMATH Keywords: linear functionals
zbMATH Keywords: field extension
zbMATH Keywords: completely decomposable modules
zbMATH Keywords: purely simple modules
MaRDI profile type: Publication
full work available at URL: 0024-3795(86)90222-3
OpenAlex ID: W4235415419
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cites work: Indecomposable modules constructed from Liouville numbers
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Wikidata QID: Q126377502
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description: scientific article; zbMATH DE number 3948525