Endomorphism rings with different rank-entropy supports.
type: Item
instance of: scholarly article
zbMATH Open document ID: 1282.20062
author: Rüdiger Göbel
author: Luigi Salce
published in: The Quarterly Journal of Mathematics
publication date: 2012-06-12T00:00:00Z
Mathematics Subject Classification ID: 20K20
Mathematics Subject Classification ID: 20K30
Mathematics Subject Classification ID: 20K45
Mathematics Subject Classification ID: 37A35
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 16W80
zbMATH DE Number: 6045450
zbMATH Keywords: torsion-free Abelian groups
zbMATH Keywords: endomorphism rings
zbMATH Keywords: rank entropy
zbMATH Keywords: \(\aleph_1\)-free Abelian groups
zbMATH Keywords: topological rings
MaRDI profile type: Publication
full work available at URL: haq048
OpenAlex ID: W2121058398
DOI: 10.1093/QMATH/HAQ048
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title: Endomorphism rings with different rank-entropy supports.
review text: This paper unites three significant areas of algebra. One is the algebraic rank entropy of torsion-free Abelian groups, that is, the chaotic behaviour of endomorphisms acting on finitely generated subgroups; the second is the structure of rings admitting different topologies; and the third is the realization of topological rings as the endomorphism rings of Abelian groups.NEWLINENEWLINE The main result is the construction of non-isomorphic \(\aleph_1\)-free groups whose endomorphism rings are algebraically, but not topologically isomorphic, one having rank entropy zero and the other rank entropy infinity. The authors also show that if two such groups have algebraically isomorphic endomorphism rings, then with respect to the finite topology, they must have the same rank entropy.
reviewed by: Phillip Schultz
description: scientific article; zbMATH DE number 6045450