This volume is an introduction to the theory of modules over algebras based on the courses given by the author. The content is entirely classical, and the topics discussed in the book can be distributed into three main parts.
The first part includes Chapters I-V, starts with the definition of an algebra over a commutative ring \(K\), and presents all the basics on categories of modules over algebras, the Hom functor, tensor products, projective and injective modules.
The second part, consisting of Chapters VI-VII, is devoted to the fundamental structure theorems. Among the landmarks, one should mention the theorems of Jordan-Hölder, Wedderburn-Artin, Maschke, Remak-Krull-Schmidt-Azumaya and Morita.
The last part, consisting of Chapters IX-XII, is an introduction to homological algebra. The Ext and Tor functors are introduced, and then the reader will find a short discussion of homological dimensions of modules and algebras, Hochschild cohomology and separable algebras, hereditary, tensor and self-injective algebras.
The presentation is very clear and careful, and all the statements have comprehensive proofs. Every chapter is accompanied by numerous and well selected exercises. This volume is definitely a very useful addition to the literature and I warmly recommend it to students who are planning to study later the representation theory and cohomology of algebras. This book will be an excellent preparation for the reading of more advanced texts. Instructors will also find here a good material for their lectures.