Abelian groups with one \(\tau\)-adic relation
type: Item
instance of: scholarly article
title: Abelian groups with one \(\tau\)-adic relation
zbMATH Open document ID: 0699.20043
DOI: 10.1007/BF01980609
published in: Algebra and Logic
publication date: 1989-01-01T00:00:00Z
review text: Let \(\tau\) be a type and \(Q_{\tau}\) be the algebra of \(\tau\)-adic numbers. If A is a torsionfree group of finite rank r and \(\{x_ 1,...,x_ r\}\) is any maximal linearly independent subset of A then the natural homomorphism \(A\to A_{\tau}\) of A into its \(\tau\)-adic completion \(A_{\tau}\) determines the submodule \[ \delta (A)=\{(\alpha_ 1,...,\alpha_ r)\in Q^ r_{\tau}| \quad \alpha_ 1x_ 1+...+\alpha_ rx_ r=0\quad in\quad A_{\tau}\} \] of \(\tau\)-relations in \(Q^ r_{\tau}\). The paper studies the properties of the class \({\mathcal F}\) of finite rank torsionfree groups A such that the module \(\delta(A)\) is cyclic. Among other results the groups from \({\mathcal F}\) are characterized up to quasi-isomorphism (Th.3.7) and pure subgroups of groups from \({\mathcal F}\) together with the corresponding factor-groups are described ({\S}{\S} 4, 5).
reviewed by: Ladislav Bican
Mathematics Subject Classification ID: 20K15
Mathematics Subject Classification ID: 20K27
Mathematics Subject Classification ID: 20K30
zbMATH DE Number: 4146264
zbMATH Keywords: type
zbMATH Keywords: \(\tau\)-adic numbers
zbMATH Keywords: torsionfree group of finite rank
zbMATH Keywords: \(\tau\)-adic completion
zbMATH Keywords: quasi-isomorphism
zbMATH Keywords: pure subgroups
author: Alexander A. Fomin
MaRDI profile type: Publication
cites work: Torsion-free rings
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cites work: Q3778926
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cites work: Torsion free groups of rank two
cites work: Typesets and cotypesets of rank-2 torsion free Abelian groups
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description: scientific article; zbMATH DE number 4146264