On the endomorphism rings of abelian groups and their Jacobson radical

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 (EndA, T L) is locally compact, where TL 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 for elementary countable p-groups A.