On the endomorphism rings of abelian groups and their Jacobson radical

Abstract

We give a characterization of those abelian groups which are direct sums of cyclic groups and the Jacobson radicals of their endomorphism rings are closed. A complete characterization of p-groups A for which \(({\textit{EndA}},\mathcal T_L)\) is locally compact, where \(\mathcal T_L\) is the Liebert topology on \({\textit{EndA}}\) is given. We prove that if A is a countable elementary p-group then \({\textit{EndA}}\) has a non-admissible ring topology. To every functorial topology on A a right bounded ring topology on \({\textit{EndA}}\) is attached. By using this topology we construct on \({\textit{EndA}}\) a non-metrizable and non-admissibe ring topology for elementary countable p-groups A.