  It is proved that if A_p is a countable elementary abelian p-group, then: (i)
The ring End(A_p) does not admit a nondiscrete locally compact ring topology.
(ii) Under (CH) the simple ring End(A_p)/I, where I is the ideal of End(A_p)
consisting of all endomorphisms with finite images, does not admit a
nondiscrete locally compact ring topology. (iii) The finite topology on
End(A_p) is the only second metrizable ring topology on it. Moreover, a
characterization of completely simple endomorphism rings of the endomorphism
rings of modules over commutative rings is also obtained.
