Two problems on \(\aleph _ 0\)-indecomposable Abelian groups
type: Item
instance of: scholarly article
title: Two problems on \(\aleph _ 0\)-indecomposable Abelian groups
zbMATH Open document ID: 0597.20049
DOI: 10.1016/0021-8693(86)90033-5
author: S. Shelah
author: Alexander Soifer
published in: Journal of Algebra
publication date: 1986-01-01T00:00:00Z
review text: An abelian group G is \(\aleph_ 0\)-indecomposable if G cannot be written as a direct sum of \(\aleph_ 0\) non-zero groups. A convenient but interesting reformulation of the \(\aleph_ 0\)-decomposability for a reduced countable mixed group G with finite torsion-free rank is that every finite subset can be imbedded in a summand of finite index. Using this the authors show that a countable reduced torsion group T is the torsion part of an \(\aleph_ 0\)-indecomposable mixed group G with torsion-free rank n exactly when \(T=F\oplus H\), where F is finite, and for each prime p and positive integer k, \(f^ b_ k(A)+f^ b_{k+1}(A)\leq n\) where \(f^ b_ k(A)\) is the kth Ulm-Kaplansky invariant of the p-component \(H_ p\).
Mathematics Subject Classification ID: 20K25
Mathematics Subject Classification ID: 20K21
Mathematics Subject Classification ID: 20K27
Mathematics Subject Classification ID: 20K99
zbMATH DE Number: 3961974
zbMATH Keywords: \(\aleph _ 0\)-decomposability
zbMATH Keywords: summand of finite index
zbMATH Keywords: countable reduced torsion group
zbMATH Keywords: \(\aleph _ 0\)-indecomposable mixed group
zbMATH Keywords: Ulm- Kaplansky invariant
Wikidata QID: Q106008766
reviewed by: Kulumani M. Rangaswamy
MaRDI profile type: Publication
full work available at URL: 0021-8693(86)90033-5
OpenAlex ID: W2002354909
cites work: Q5609457
cites work: Q5534953
cites work: Q3317294
cites work: Criteria for the existence of irreducible systems of generators for abelian groups
cites work: Q3681036
description: scientific article; zbMATH DE number 3961974