Completely simple endomorphism rings of modules

TL;DR

This paper investigates the topological properties of endomorphism rings of countable elementary abelian p-groups, establishing their unique topological structures and characterizations of completely simple endomorphism rings.

Contribution

It provides new results on the topological structure and characterization of endomorphism rings of modules over commutative rings, especially for elementary abelian p-groups.

Findings

End(A_p) admits no nondiscrete locally compact ring topology.

Under CH, the simple ring End(A_p)/I also admits no such topology.

The finite topology is the only second metrizable ring topology on End(A_p).

Abstract

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.