The first edition of the author’s comprehensive textbook “Advanced modern algebra” appeared in 2002, published at that time by Prentice Hall/Pearson Education [[Upper Saddle River, NJ: xv, 1012 p., append. 27 p. (2002; Zbl 0997.00001; ME 2011b.00704)]. The book under review is the second edition of this popular primer of graduate abstract algebra,which appears now as a volume in the renowned textbook series “Graduate studies in mathematics” of the American Mathematical Society (AMS). As for this second edition at hand, not only the publisher has changed, but also the text itself appears in thoroughly revised form. Actually, the author has reorganized and rewritten large parts of book in order to achieve substantial improvements regarding its depth, versatility, and general expediency. Altogether, the whole exposition has been tightened up for the benefit of the more advanced parts of the material, including new important topics, concepts, methods, and applications. More precisely, the following major differences with the first edition of the book are particularly noteworthy: 1) Instead of devoting the first three chapters to a review of some elementary material from undergraduate algebra (as it was done in the first edition), the author usually refers to his foregoing textbook “A first course in abstract algebra” [Upper Saddle River, NJ: Prentice Hall. xi, 265 p. (1996; Zbl 0847.00004; ME 1997a.00453)] as for those prerequisites. Accordingly, the original first chapter entitled “Things past” has been omitted in the present second edition of the book, which now begins with a discussion of algebraic equations of low degree, permutations, and basic group theory. 2) Noncommutative rings are now introduced much earlier in the book, together with general module theory, categories, functors, and the related aspects of classical homological algebra. 3) The former Chapter 8, entitled “Algebras” in the first edition, has been completely rewritten, appearing now as Chapter 7 with the headline “Representation theory”. This chapter also discusses some new topics such as division rings, Brauer groups, abelian categories, and module categories. 4) In the chapter on homology (now Chapter 9), the section on elementary algebraic $K$-theory has been thoroughly revised and enlarged, thereby focussing on the Grothendieck groups $K_0$ and $G_0$. 5) Some other new aspects taken up here are Galois theory for infinite field extensions and the normal basis theorem for rings of algebraic integers. 6) The numerous misprints and inaccuracies in the first edition of the book have been corrected, and the overall presentation has been improved wherever appropriate. As a result of the different organization of the material, the present second edition of the author’s excellent, utmost versatile and sweeping graduate text on modern abstract algebra comes now with ten chapters which are arranged as follows: 1. Groups I; 2. Commutative Rings I; 3. Galois Theory; 4. Groups II; 5. Commutative Rings II; 6. (General) Rings; 7. Representation Theory; 8. Advanced linear Algebra; 9. Homology; 10. Commutative Rings III. Spirit, philosophy, didactic principles, and the essential contents of these (old and new) chapters have been left totally unchanged, apart from the changes and novelties mentioned above, and therefore we may in every respect refer to the detailed review of the first edition of this masterly textbook. Now as before, the author’s wonderful introduction to the vast field of advanced modern algebra should be seen as one of the great standard references in the relevant textbook literature, standing out by its many unique, appraised features.

Reviewer:

Werner Kleinert (Berlin)