  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 $A$.
