Commutator invariant subgroups of Abelian groups.
type: Item
instance of: scholarly article
title: Commutator invariant subgroups of Abelian groups.
zbMATH Open document ID: 1210.20053
published in: Siberian Mathematical Journal
publication date: 2010-12-08T00:00:00Z
review text: The commutator \([\varphi,\psi]\) of two elements of a ring is the element \(\varphi\psi-\psi\varphi\). A subgroup \(H\) of an Abelian group \(A\) is commutator invariant if \([\varphi,\psi]H\subseteq H\) for all commutators in the endomorphism ring of \(A\). The purpose of this paper is to describe the commutator invariant subgroups of an Abelian group. In particular, the author shows that the commutator invariant subgroups of separable and algebraically compact torsion-free Abelian groups are fully invariant. The \(E\)-center of \(A\) is the subgroup annihilated by all commutators, and the \(E\)-commutant is the subgroup generated by the images of all commutators. The author describes the \(E\)-centers and the \(E\)-commutants of several notable classes of Abelian groups.
reviewed by: Phillip Schultz
Mathematics Subject Classification ID: 20K30
Mathematics Subject Classification ID: 20K27
zbMATH DE Number: 5824094
zbMATH Keywords: endomorphism rings
zbMATH Keywords: fully invariant subgroups
zbMATH Keywords: commutators of endomorphisms
zbMATH Keywords: \(E\)-center
zbMATH Keywords: \(E\)-commutant
author: Andrey R. Chekhlov
MaRDI profile type: Publication
full work available at URL: s11202-010-0092-6
OpenAlex ID: W2058547149
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DOI: 10.1007/S11202-010-0092-6
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description: scientific article; zbMATH DE number 5824094