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.
