We introduce here the notations and terminology we use on algebras and modules, and we briefly recall some of the basic facts from module theory. Examples of algebras, modules, and functors are presented. We introduce the notions of the (Jacobson) radical of an algebra and of a module; the notions of semisimple module, projective cover, injective envelope, the socle and the top of a module, local algebra, and primitive idempotent. We also collect basic facts from the module theory of finite dimensional K-algebras. In this chapter we present complete proofs of most of the results, except for a few classical theorems. In these cases the reader is referred to the following textbooks on this subject, and.
Throughout, we freely use the basic notation and facts on categories and functors introduced in the Appendix.
The reader interested mainly in linear representations of quivers and path algebras or familiar with elementary facts on rings and modules can skip this chapter and begin with Chapter II.
For the sake of simplicity of presentation, we always suppose that K is an algebraically closed field and that an algebra means a finite dimensional K-algebra, unless otherwise specified.
Algebras
By a ring, we mean a triple (A, +, ·) consisting of a set A, two binary operations: addition + : A × A → A, (a, b) ↦ a + b; multiplication · : A × A → A, (a, b) ↦ ab, such that (A, +) is an abelian group, with zero element 0 ∈ A, and the following conditions are satisfied:
(i) (ab)c = a(bc)
(ii) a(b + c) = ab + ac and (b + c)a = ba + ca
for all a, b, c ∈ A.