While we are assuming that the reader is familiar with general concepts of ring theory, such as the radical of a ring, and of module theory, such as projective, injective and simple modules, we are not assuming that the reader, except for semisimple modules and semisimple rings, is necessarily familiar with the special features of the structure of artin algebras and their finitely generated modules. This chapter is devoted to presenting background material valid for left artin rings, and the next chapter deals with special features of artin algebras. All rings considered in this book will be assumed to have an identity and all modules are unitary, and unless otherwise stated all modules are left modules.
We start with a discussion of finite length modules over arbitrary rings. After proving the Jordan–Hölder theorem, we introduce the notions of right minimal morphisms and left minimal morphisms and show their relationship to arbitrary morphisms between finite length modules. When applied to finitely generated modules over left artin rings, these results give the existence of projective covers which in turn gives the structure theorem for projective modules as well as the theory of idempotents in left artin rings. We also include some results from homological algebra which we will need in this book.
Finite length modules
In this section we introduce the composition series and composition factors for modules of finite length. We prove the Jordan–Hölder theorem and give an interpretation of it in terms of Grothendieck groups.