Construction of dual modules using Martin's axiom.
type: Item
instance of: scholarly article
title: Construction of dual modules using Martin's axiom.
zbMATH Open document ID: 1170.16005
author: Rüdiger Göbel
author: Sebastian Pokutta
published in: Journal of Algebra
publication date: 2008-11-12T00:00:00Z
review text: It is a challenging open problem to classify all Abelian groups \(G\) that are dual groups, which means \(G=H^*=\Hom(H,\mathbb{Z})\) for some Abelian group \(H\). The authors attack this problem using additional set-theory by applying Martin's axiom. Working in the category of \(R\)-modules over countable principal ideal domains \(R\) the authors are able to represent large classes of modules that are submodules of \(R^\omega\) as dual modules. The idea is to reduce the problem of solving infinite systems of linear equations over \(R\) to finite ones using Martin's axiom. The obtained modules are sandwiched between \(R^{(\omega)}\) and its closure inside \(R^\omega\). Going one step further the authors are also able to demand even more on the groups by requiring that the endomorphism ring is minimal, i.e. \(\text{End}(G)=R\oplus\text{Fin}(G)\). The paper is well-written and can certainly be recommended to anyone who is interested in dual groups and the use of Martin's axiom.
Mathematics Subject Classification ID: 16D80
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 16B70
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 20K30
Mathematics Subject Classification ID: 13C13
zbMATH DE Number: 5365171
zbMATH Keywords: Martin axiom
zbMATH Keywords: dual modules
zbMATH Keywords: Abelian groups
zbMATH Keywords: dual groups
zbMATH Keywords: endomorphism rings
reviewed by: Lutz Strüngmann
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2008.06.017
OpenAlex ID: W2072398417
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DOI: 10.1016/J.JALGEBRA.2008.06.017
description: scientific article; zbMATH DE number 5365171