Completely simple endomorphism rings of modules

Original · EN

It is proved that if Aₚ is a countable elementary abelian p-group, then: (i) The ring End(Aₚ) does not admit a nondiscrete locally compact ring topology. (ii) Under (CH) the simple ring End(Aₚ)/I, where I is the ideal of End(Aₚ) consisting of all endomorphisms with finite images, does not admit a nondiscrete locally compact ring topology. (iii) The finite topology on End(Aₚ) 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.