Quasi-decomposition of abelian groups and Baer's lemma
type: Item
instance of: scholarly article
title: Quasi-decomposition of abelian groups and Baer's lemma
zbMATH Open document ID: 0798.20049
published in: Rocky Mountain Journal of Mathematics
publication date: 1993-06-29T00:00:00Z
review text: Let \(A\) be a finite rank torsion-free abelian group. An abelian group \(G\) is called quasi-\(A\)-projective if \(G\) is quasi-isomorphic to a direct summand of a finite direct sum of copies of \(A\). For any group \(C\), \(S_ A(C)\) is the subgroup of \(C\) generated by \(\{\varphi(A):\varphi\in \text{Hom}(A,C)\}\). The abelian group \(A\) is said to have the quasi-Baer- splitting property if every exact sequence \(0 \to B @>\alpha>> C @>\beta>> G \to 0\) where \(G\) is quasi-\(A\)-projective and \(C\) is quasi equal to \(\alpha(B) + S_ A(C)\) is quasi-splitting. The author gives necessary and sufficient conditions for \(A\) to have the quasi-Baer- splitting property in terms of the strongly indecomposable summands \(A_ 1,\dots,A_ n\) that show up in a fixed quasi-decomposition of \(A\): These groups \(A_ i\) must have incomparable socles, which properly is defined in terms of the socles \(S_{A_ j}(X)\) of subgroups \(X\) of \(A_ i\) for various \(i\) and \(j\). In particular, an almost completely decomposable group \(A\) will have the quasi-Baer-splitting property exactly when all its critical types are pair-wise incomparable.
Mathematics Subject Classification ID: 20K15
Mathematics Subject Classification ID: 20K40
Mathematics Subject Classification ID: 20K25
Mathematics Subject Classification ID: 20K30
Mathematics Subject Classification ID: 20K27
zbMATH DE Number: 219330
zbMATH Keywords: finite rank torsion-free abelian group
zbMATH Keywords: quasi-\(A\)-projective
zbMATH Keywords: direct summand
zbMATH Keywords: finite direct sum
zbMATH Keywords: quasi-Baer-splitting property
zbMATH Keywords: quasi- splitting
zbMATH Keywords: strongly indecomposable summands
zbMATH Keywords: quasi-decomposition
zbMATH Keywords: socles
zbMATH Keywords: almost completely decomposable group
zbMATH Keywords: types
author: Ulrich F. Albrecht
reviewed by: Kulumani M. Rangaswamy
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181072651
OpenAlex ID: W2087973301
Wikidata QID: Q125052918
cites work: Q3663525
cites work: Baer's Lemma and Fuchs' Problem 84a
cites work: Faithful Abelian Groups of Infinite Rank
cites work: Endomorphism rings of faithfully flat Abelian groups
cites work: Finite rank torsion free Abelian groups and rings
cites work: Endomorphism Rings and Direct Sums of Torsion Free Abelian Groups
cites work: Abelian groups, \(A\), such that \(\Hom(A,-)\) preserves direct sums of copies of \(A\)
cites work: Abelian groups without elements of finite order
cites work: Q5609457
cites work: On direct decompositions of torsionfree Abelian groups
cites work: Expansion of the Confluent Hypergeometric Function in Series of Bessel Functions
cites work: A Note on Torsion Free Abelian Groups of Infinite Rank
cites work: Q3882615
DOI: 10.1216/RMJM/1181072651
description: scientific article; zbMATH DE number 219330