On the endomorphism rings of abelian groups and their Jacobson radical

TL;DR

This paper characterizes abelian groups with certain properties of their endomorphism rings, explores topologies on these rings, and constructs examples of non-metrizable and non-admissible ring topologies.

Contribution

It provides a complete characterization of abelian groups with closed Jacobson radicals and analyzes topologies on endomorphism rings of p-groups, introducing new non-metrizable topologies.

Findings

Characterization of abelian groups as direct sums of cyclic groups with closed Jacobson radical.

Conditions for local compactness of endomorphism rings of p-groups under Liebert topology.

Existence of non-metrizable, non-admissible ring topologies on endomorphism rings of countable elementary p-groups.

Abstract

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