4. Algebras and Modules

This chapter is an introduction to k-algebras and their modules, where k is an algebraically closed field. Since every algebra is a ring, we will often use certain notions from ring theory, like ideals and radicals. We introduce these notions in the first section. In the second and the third section we define k-algebras and their modules and present examples and basic properties. In the fourth section, we study the direct sum decomposition of a k-algebra (as a module over itself) determined by a choice of a complete set of primitive orthogonal idempotents \(e_{1},\ldots,e_{n}\). For the path algebra of a quiver Q, we are already familiar with this construction, namely the idempotent e _i corresponds to the constant path at the vertex i in Q and the direct sum decomposition of the algebra corresponds to the direct sum of all indecomposable projective representations P(i). In the fifth section, we prove a useful criterion for the indecomposability of a module M. In fact, we show that M is indecomposable if and only if the algebra of all endomorphisms of M is a local algebra.