A generalization of Baer's splitting problem on Abelian groups
type: Item
instance of: scholarly article
title: A generalization of Baer's splitting problem on Abelian groups
zbMATH Open document ID: 0678.20034
DOI: 10.1007/BF01170940
published in: Manuscripta Mathematica
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155419
review text: The paper treats a generalization of Baer's problem of describing all abelian groups A satisfying \(Ext(A,T)=0\) for all abelian torsion groups T. Here the given class of abelian groups is the full subcategory \({\mathcal C}_ G\) of the category \({\mathcal A}b\) of all abelian groups which is generated by an abelian group G. All abelian groups A with \(Ext(A,H)=0\) for all \(H\in {\mathcal C}_ G\) form another subcategory \({\mathcal A}_ G\) of \({\mathcal A}b\). The problem of characterizing the members of \({\mathcal A}_ G\) is reduced to torsion-free abelian groups. It turns out that a torsion- free abelian group A is in \({\mathcal A}_ G\) exactly if it is the union of a continuous well-ordered ascending chain of pure subgroups \(0=C_ 0<C_ 1<...<C_{\nu}<...<C_{\sigma}=A\) for some ordinal \(\sigma\) whose rank 1 factors \(B_{\nu}=C_{\nu +1}/C_{\nu}\) satisfy the conditions (i) \(pB_{\nu}\neq B_{\nu}\) whenever \(p^{k+1}G\neq p^ kG\) for all \(k\geq 0\) and all primes p; (ii) the type t of \(B_{\nu}\) can be realized by a characteristic \((...,k_ p,...)\) where \(k_ p=0\) for almost all primes p with pG\(\neq G\).
Mathematics Subject Classification ID: 20K40
Mathematics Subject Classification ID: 20K35
Mathematics Subject Classification ID: 20K20
zbMATH DE Number: 4110949
zbMATH Keywords: Ext
zbMATH Keywords: Hom
zbMATH Keywords: Baer's problem
zbMATH Keywords: category
zbMATH Keywords: torsion-free abelian groups
zbMATH Keywords: continuous well-ordered ascending chain of pure subgroups
zbMATH Keywords: type
zbMATH Keywords: characteristic
author: László Fuchs
reviewed by: Huberta Lausch
MaRDI profile type: Publication
cites work: Baer modules over valuation domains
cites work: Abelian group extensions and the axiom of constructibility
cites work: Q5609457
cites work: Q3778162
cites work: New aspects for two classical theorems on torsion splitting*
cites work: A Solution to the Splitting Mixed Group Problem of Baer
cites work: Incompactness in regular cardinals
OpenAlex ID: W2041350276
description: scientific article; zbMATH DE number 4110949