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Advanced modern algebra. Part 2. 3rd edition. (English) Zbl 1390.00002

Graduate Studies in Mathematics 180. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-2311-7/hbk; 978-1-4704-4170-8/ebook). ix, 558 p. (2017).
This graduate algebra book, written by the well-know mathematical writer Joseph Rotman, is the second part of the third edition of Advanced Modern Algebra. The author decided to split the third edition in two volumes in order to use as a textbook for different graduate courses. The first edition was published in 2002 [Zbl 0997.00001], the second in 2010 [Zbl 1206.00007] and Part I in 2015 [Zbl 1332.00008].
The material of this book is divided in five chapters. Chapter I is devoted to group theory and it presents more advanced topics in groups that the ones presented in part I. These include group actions, Sylow theorems, soluble and nilpotent groups, simplicity of \(\mathrm{PSL}(n,q)\) (two different proofs), the Nielsen-Schreier theorem and the Baer-Levi proof of the last theorem.
The next chapter deals with representation theory. After recalling some properties of chain conditions on modules and rings from Part I, the Jacobson radical is studied. Semisimple rings are introduced and Maschke, Wedderburn, Wedderburn-Artin and Hopkins-Levitzki theorems are proved. After a short introduction to Lie algebras, character theory is considered. This theory is developed and the Burnside theorem on soluble groups and the Frobenius theorem ared showed. The last section of this chapter contains an introduction to division ring and Brauer group.
Chapter C-3 is titled “Homology” and includes semidirect products, general extensions and cohomology, the Schreier theorem, the Schur-Zassenhaus lemma, homology functors, derived functors, Ext and Tor functors, cohomology of grups, crossed products and an introduction to spectral sequences.
In Chapter C-4, the author returns to category theory, introduced in Chapter B-4 of Part I. After studying additive and abelian categories, sheaves are considered from the two equivalent approaches, g-sheaves and special presheaves and a short introduction to cohomology of sheaves is included. The Gabriel-Mitchell theorem on characterization of abelian categories equivalent to a module category and the theorem on the existence of adjoint functor for modules are two important results included in this chapter. In the last section, a short introduction to algebraic \(K\)-theory is considered.
The final chapter of this book deals with commutative rings. In this part III of commutative rings, homological methods are applied to its study. The first main idea is the local-global principle. This idea is illustrated with abelian groups and next with commutative localization theory. Dedeking rings is the content of the following section. This study includes integrality, algebraic integers, several characterizations of Dedeking rings and the structure of finitelely generated modules over Dedeking domains. The final aim of this chapter is to give the homological characterization of local regular rings by Serre-Auslander-Buchsbaum, a key cornerstone of homological theory . In the way, homological dimension of modules and rings are introduced and its main properties obtained. The Hilbert theorem of syzygies (the global dimension of polynomial rings over a field with \(n\) indeterminates is \(n\)) and the Auslander-Buchsbaum theorem (every regular local ring is a unique factorization domain) are shown.
This excellent advanced book on algebra will become a classical among the recent modern algebra books. It shares with Part I and the preceding two editions the same accurate didactic style of the author. I would recommend it as a reference book for experts interested in these topics of algebra and as a textbook for graduate students. In fact, there are several possibilities to teach graduate courses from this text. Every chapter of the book includes a selected numbers of exercises and some hints for more advances topics.

MSC:

00A05 Mathematics in general
12-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to field theory
18-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to category theory
13-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra
16-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to associative rings and algebras
20-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory
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