Semisimple Modules and Semisimple Algebras

Abstract

Since simple modules are the ‘building blocks’ for arbitrary finite-dimensional modules, one would like to understand how modules are built up from simple modules. In this chapter we study modules which are direct sums of simple modules, this leads to the theory of semisimple modules. For some algebras, every module is a direct sum of simple modules. These algebras are said to be semisimple, and they have important applications. In this chapter, as an exception, we deal with arbitrary direct sums of modules. We view this as an opportunity to prove a result which does not need finiteness assumptions. The only new tool necessary is Zorn’s lemma.