Nonstandard uniserial modules over valuation domains
type: Item
instance of: scholarly article
title: Nonstandard uniserial modules over valuation domains
zbMATH Open document ID: 0679.13003
DOI: 10.1007/BF03322380
author: Berthold Franzen
author: Rüdiger Göbel
published in: Results in Mathematics
publication date: 1987-01-01T00:00:00Z
review text: A commutative ring with identity is a valuation ring if its lattice of ideals is totally ordered by inclusion, it is a valuation domain if in addition it has no zero divisors. Clearly epic images of valuation domains are valuation rings. Some time ago, I. Kaplansky raised the question if all valuation rings are of this form. Only quite recently a partial answer could be given by \textit{L. Fuchs} and \textit{L. Salce} in their book ``Modules over valuation domains'' (1985; Zbl 0578.13004): In the constructible universe they exhibited a valuation ring of cardinality \(\omega_ 1\), which is not an epic image of a valuation domain. This was carried out by applying Jensen's combinatorial principle \(\diamond_{\omega_ 1}.\) The purpose of this note is to show that a set theoretic assumption weaker than the continuum hypothesis suffices, namely \(2^{\aleph_ 0}<2^{\aleph_ 1}\).
Mathematics Subject Classification ID: 13A18
Mathematics Subject Classification ID: 13L05
Mathematics Subject Classification ID: 03H15
zbMATH DE Number: 4112731
zbMATH Keywords: nonstandard uniserial modules
zbMATH Keywords: valuation domain
MaRDI profile type: Publication
cites work: Fully rigid systems of modules
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cites work: Q3699812
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description: scientific article; zbMATH DE number 4112731