The quasi-Baer-splitting property for mixed Abelian groups.
type: Item
instance of: scholarly article
title: The quasi-Baer-splitting property for mixed Abelian groups.
zbMATH Open document ID: 1060.20045
author: Simion Sorin Breaz
published in: Journal of Pure and Applied Algebra
publication date: 2004-08-19T00:00:00Z
review text: Quasi decomposition theory for Abelian groups has been of great interest over the last decades. Hence the Abelian category \(\mathbb{Q} Ab\) with all Abelian groups as objects and morphisms \(\mathbb{Q}\otimes\Hom(A,C)\) as morphisms between Abelian groups \(A\) and \(C\) has been considered. Many concepts in the theory of Abelian groups have a quasi-analogue, i.e. a similar concept in the quasi category. The author of this paper studies the quasi Baer splitting property. An Abelian group \(C\) is called `almost \(A\)-generated' for some Abelian group \(A\) if there exists an epimorphism \(A^{(I)}\to C\) in \(\mathbb{Q} Ab\). Moreover, \(C\) is called `almost \(A\)-projective' if it is isomorphic to a quasi-summand of a direct sum of copies of \(A\). Accordingly, the Abelian group \(A\) has the `quasi Baer splitting property' if every epimorphism \(C\to G\to 0\), with \(C\) an almost \(A\)-generated group and \(G\) an almost \(A\)-projective group, quasi-splits. The author proves various characterizations of self-small groups which have the quasi Baer splitting property. Further results are shown, e.g. a self-small group \(A\) is almost \(E\)-flat and has the quasi Baer splitting property if and only if the class of almost \(A\)-adstatic modules is closed with respect to submodules and if and only if \(A\) is almost-projective with respect to the class of almost \(A\)-static groups.
Mathematics Subject Classification ID: 20K21
Mathematics Subject Classification ID: 20K30
Mathematics Subject Classification ID: 20K25
Mathematics Subject Classification ID: 20K40
zbMATH DE Number: 2093439
zbMATH Keywords: Baer splitting property
zbMATH Keywords: self-small Abelian groups
zbMATH Keywords: quasi-decompositions
zbMATH Keywords: quasi-summands
zbMATH Keywords: endomorphism rings
reviewed by: Lutz Strüngmann
MaRDI profile type: Publication
full work available at URL: j.jpaa.2003.12.008
OpenAlex ID: W1997898030
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DOI: 10.1016/J.JPAA.2003.12.008
description: scientific article; zbMATH DE number 2093439