Transitivity and full transitivity for nontorsion modules
type: Item
instance of: scholarly article
title: Transitivity and full transitivity for nontorsion modules
zbMATH Open document ID: 0890.20039
author: Steve T. Files
published in: Journal of Algebra
publication date: 1998-04-21T00:00:00Z
review text: Recall that an \(R\)-module \(M\) is (fully) transitive if for all \(x,y\in M\) with \(U_M(x)=U_M(y)\) (\(U_M(x)\leq U_M(y)\)) there exists an \(R\)-automorphism (\(R\)-endomorphism) \(\phi\) of \(M\) such that \(x\phi=y\), where \(U_M(x)\) denotes the height sequence of \(x\) in \(M\). In the first part of the paper the author studies weakened notions of transitivity and full transitivity which are specially suited for use with mixed modules. The second part is devoted to the connection between transitivity and full transitivity for mixed modules. It is shown that these two notions are equivalent for the class of Warfield modules (direct summands of simply presented modules). Another result asserts that a completely decomposable module is fully transitive if and only if its square is transitive. For torsion modules this was shown by the author and \textit{B. Goldsmith} [in Proc. Am. Math. Soc. 126, No. 6, 1605-1610 (1998)]. The author conjectures that all modules obey such a law.
Mathematics Subject Classification ID: 20K21
Mathematics Subject Classification ID: 20K30
Mathematics Subject Classification ID: 16D70
zbMATH DE Number: 1104109
zbMATH Keywords: \(p\)-local Abelian groups
zbMATH Keywords: height sequences
zbMATH Keywords: fully transitive Abelian groups
zbMATH Keywords: completely decomposable modules
zbMATH Keywords: Warfield modules
zbMATH Keywords: mixed modules
zbMATH Keywords: simply presented modules
MaRDI profile type: Publication
full work available at URL: jabr.1997.7062
OpenAlex ID: W2093058201
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DOI: 10.1006/JABR.1997.7062
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description: scientific article; zbMATH DE number 1104109