Separable torsion-free Abelian \(E^*\)-groups
type: Item
instance of: scholarly article
title: Separable torsion-free Abelian \(E^*\)-groups
zbMATH Open document ID: 0927.20038
DOI: 10.1016/S0022-4049(97)00195-3
published in: Journal of Pure and Applied Algebra
publication date: 1999-10-03T00:00:00Z
review text: The first half of this paper characterizes the torsion-free separable abelian groups \(G\) whose endomorphism semigroup \(E(G)^*\) admits a unique addition; that is, the endomorphism ring \(E(G)\) is isomorphic to any ring \(S\) for which \(E(G)^*\) is isomorphic to \(S^*\). It is shown that a necessary and sufficient condition is that the group \(G\) is semiconnected, that is, \(G\) has rank \(>1\) and a complementary summand of any rank 1 summand \(A\) has a rank 1 summand comparable with \(A\). The second half concerns \(E^*\)-groups, that is, torsion-free separable groups \(G\) such that for all rings \(R\) with \(R^*\) isomorphic to \(E(G)^*\), there is an abelian group \(H\) with \(E(H)\) isomorphic to \(R\). It is shown that a rank 1 group is an \(E^*\)-group if and only if it is divisible by only finitely many primes; and in general, \(G\) is an \(E^*\)-group if and only if every rank 1 summand which is not semiconnected is divisible by only finitely many primes.
reviewed by: Phillip Schultz
Mathematics Subject Classification ID: 20K20
Mathematics Subject Classification ID: 20K30
Mathematics Subject Classification ID: 16S50
zbMATH DE Number: 1308261
zbMATH Keywords: unique addition rings
zbMATH Keywords: semigroup isomorphisms
zbMATH Keywords: torsion-free separable Abelian groups
zbMATH Keywords: endomorphism semigroups
zbMATH Keywords: endomorphism rings
zbMATH Keywords: rank \(1\) summands
author: Oleg V. Lyubimtsev
author: A. M. Sebel'din
author: Charles Vinsonhaler
MaRDI profile type: Publication
cites work: Q3858470
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description: scientific article; zbMATH DE number 1308261