Countable \(\aleph _ 0\)-indecomposable mixed Abelian groups of finite torsion-free rank
type: Item
instance of: scholarly article
title: Countable \(\aleph _ 0\)-indecomposable mixed Abelian groups of finite torsion-free rank
zbMATH Open document ID: 0586.20029
DOI: 10.1016/0021-8693(86)90085-2
author: S. Shelah
author: Alexander Soifer
published in: Journal of Algebra
publication date: 1986-01-01T00:00:00Z
review text: A group is \(\aleph_ 0\)-indecomposable if it cannot be decomposed into a direct sum of \(\aleph_ 0\) non-zero summands. In this paper the authors find necessary and sufficient conditions for an extension of a countable reduced torsion abelian group T by a finite rank torsion-free abelian group R to be \(\aleph_ 0\)-indecomposable. These conditions are remarkably simple, and depend only on T and R: Let R have rank n and Richman type the matrix t. Then T has a basic subgroup which decomposes as a direct sum of \(n+1\) summands \(B_ 0,B_ 1,...,B_ n\) such that \(B_ 0\) is finite; for \(i\geq 1\), \(B_ i\) has no two adjacent non-zero Ulm-Kaplansky invariants; and for \(i\geq 1\) and all primes p, \([r_{pi}]\leq t\), where \(r_{pi}\) is the p-rank of \(B_ i\).
Mathematics Subject Classification ID: 20K21
Mathematics Subject Classification ID: 20K25
Mathematics Subject Classification ID: 20K99
zbMATH DE Number: 3939591
zbMATH Keywords: extension
zbMATH Keywords: reduced torsion abelian group
zbMATH Keywords: finite rank torsion-free abelian group
zbMATH Keywords: \(\aleph _ 0\)-indecomposable
zbMATH Keywords: Richman type
zbMATH Keywords: basic subgroup
zbMATH Keywords: direct sum
zbMATH Keywords: Ulm-Kaplansky invariants
reviewed by: Phillip Schultz
MaRDI profile type: Publication
full work available at URL: 0021-8693(86)90085-2
OpenAlex ID: W2060566767
cites work: Endomorphism Rings and Direct Sums of Torsion Free Abelian Groups
cites work: Note on extensions of Abelian groups by primary groups
cites work: Q5573434
cites work: Two problems on \(\aleph _ 0\)-indecomposable Abelian groups
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description: scientific article; zbMATH DE number 3939591