On the Jacobson radical of the endomorphism ring of a homogeneous separable group.
type: Item
instance of: scholarly article
title: On the Jacobson radical of the endomorphism ring of a homogeneous separable group.
zbMATH Open document ID: 1204.20072
DOI: 10.1134/S0001434610010165
author: A. V. Budanov
published in: Mathematical Notes
publication date: 2010-09-24T00:00:00Z
review text: For every Abelian group \(G\) with the endomorphism ring \(E\), there are two ideals connected with the Jacobson radical of \(E\), \(J(E)\). These ideals are \(H(G)\), the set of all \(\alpha\in E\) such that the divisible part of \(G\) is included in the kernel of \(\alpha\) and \(\alpha\) strictly increases all finite \(p\)-heights, respectively \(F(G)\), the set of all endomorphisms of \(G\) with finite rank images. If \(G\) is completely decomposable, then \(J(E)=H(G)\cap F(G)\), while if \(G\) is homogeneous separable then \(H(G)\cap F(G)\subseteq J(G)\subseteq H(G)\). The author proves that for a homogeneous separable torsion-free group \(G\) the equality \(J(E)=H(G)\) holds if and only if \(J(E)\) is closed with respect to the finite topology of \(E(G)\) (Theorem 4), and the equality \(J(E)=H(G)\cap F(G)\) holds if and only if for every endomorphism \(\alpha\in J(E)\) there is a completely decomposable direct summand \(A\) of \(G\) such that the image of \(\alpha\) is contained in \(A\).
reviewed by: Simion Sorin Breaz
Mathematics Subject Classification ID: 20K30
Mathematics Subject Classification ID: 20K20
zbMATH DE Number: 5791028
zbMATH Keywords: Jacobson radical
zbMATH Keywords: endomorphism rings
zbMATH Keywords: homogeneous separable groups
zbMATH Keywords: torsion-free Abelian groups
MaRDI profile type: Publication
cites work: Endomorphism rings of Abelian groups
cites work: Q5673086
description: scientific article; zbMATH DE number 5791028