An application of set theory to \(\omega+n\)-totally \(p^{\omega+n}\)-projective primary Abelian groups.
type: Item
instance of: scholarly article
title: An application of set theory to \(\omega+n\)-totally \(p^{\omega+n}\)-projective primary Abelian groups.
zbMATH Open document ID: 1247.20063
author: Peter Danchev
author: Patrick W. Keef
published in: Mediterranean Journal of Mathematics
publication date: 2012-02-24T00:00:00Z
review text: Recall that a group is called \(\omega\)-totally \(\Sigma\)-cyclic if it is a primary Abelian group whose separable subgroups are all direct sums of cyclic groups. The two authors study the class of such groups and their main result is Theorem 2.6. that contains a number of equivalent descriptions of the groups in this class. As a consequence the converse of a theorem due to Megibben is obtained. As a natural generalization the authors then consider the notion of \(\omega+n\)-totally \(p^{\omega+n}\)-projective groups and translate the existence of some \(\omega+n\)-totally \(p^{\omega+n}\)-projective group that is not \(\omega\)-totally \(\Sigma\)-cyclic into a question on valuated vector spaces. This is Theorem 3.11. Finally they prove that these statements are undecidable in ZFC.
Mathematics Subject Classification ID: 20K10
Mathematics Subject Classification ID: 03E35
Mathematics Subject Classification ID: 20K25
zbMATH DE Number: 6009085
zbMATH Keywords: primary Abelian groups
zbMATH Keywords: direct sums of cyclic groups
zbMATH Keywords: valuated vector spaces
zbMATH Keywords: totally projective groups
zbMATH Keywords: undecidability in ZFC
reviewed by: Lutz Strüngmann
MaRDI profile type: Publication
full work available at URL: s00009-010-0088-2
OpenAlex ID: W2034000513
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DOI: 10.1007/S00009-010-0088-2
description: scientific article; zbMATH DE number 6009085