Introduction to ring theory / P. M. Cohn

As to the introduction to rings under review, the undergraduate version of the subject, the material is arranged in five chapters, each of which is subdivided into several sections. Chapter 1 is entitled “Basics” and gives the basics on rings, ideals, and modules over arbitrary rings. This introductory chapter also contains some basic material on fields, vector spaces, and matrices as well as an explanation of the language of categories and functors. Chapter 2 discusses linear algebras over arbitrary rings, chain conditions for modules and ideals, and treats then Artinian rings in greater detail, including Wedderburn's theorem. In this context, the author touches upon modules of finite length, the Krull-Remak-Schmidt theorem, Fitting's lemma, and concludes this chapter with an introduction to group representations and group characters. Chapter 3 deals with Noetherian rings, in general, and with various examples of them. In this general framework, the author treats polynomial rings, power series rings, unique factorization domains, Euclidean rings, principal ideal domains, modules over principal ideal domains, and rings of algebraic integers in a real-quadratic number field. Chapter 4 comes with the very general title “Ring constructions”. In seven separate sections the author discusses various topics ranging from direct products of rings, tensor products of modules and algebras, bimodules, adjoint functors, projective modules, injective modules, and some hints to higher homological algebra up to the much more specific subject of invariant-basis-number rings and projective-free rings in noncommutative ring theory. The latter topic is one of the author's special domains of research and leads the reader to the more recent developments in general ring theory. In Chapter 5, the final chapter, the author indulges his personal interests and tastes even more. He takes the reader deeper into noncommutative algebra and describes some aspects of general rings which help the reader to better understand the basic concepts. Their inclusion is indeed justified by the fact that they are usually only found in specialist accounts but, on the other hand, do not require extensive background knowledge, which makes them fit perfectly in such an introductory text. More precisely, the author discusses rings of fractions in general, skew polynomial rings, free algebras, general tensor rings, and free ideal rings. Most of these topics have only recently been developed, and they are not yet too widely known, although they are now beginning to find important applications in topology and functional analysis. Therefore it is very useful and gratifying to see these special topics included, for the first time, in an introductory textbook. This introduction to ring theory for undergraduates appears very clear and well-structured, purposeful, enlightening and didactically efficient. The arrangement of the material is very original and inspiring for teachers in the field, and the presentation as a whole is pleasantly refined, just as customary for this experienced author. (Zentralblatt)

Bibliogr. p. 223-224. Index