Study on semisimple rings and modules

Abstract

In this thesis, we characterize semisimple modules over noncommutative rings and investigate their properties. We discuss noncommutative rings and their modules based on the Wedderburn-Artin structure theorem. Focusing on the basic concept of a semisimple module, we prove that a module over a semisimple ring is again semisimple. Considering the modular law, we prove that every submodule of a semisimple module contains a simple submodule. Some characterizations of semisimple modules over associative rings are also available in this study. We study some characterizations of regular rings. We show that every semisimple module is a quasi-projective module. Establishing the structure of endomorphism rings, we prove that the endomorphism ring of a semisimple module is regular. Finally, we prove that if M is a regular module and S is an endomorphism ring, then for any α∈S, α(M) is a direct summand of M; conversely, when M is quasi-projective and α(M) is a direct summand of M for any α∈S, then M is regular.