Algebra. A Graduate Course
This textbook introduces the reader to methods of modern algebra. It is based on a graduate algebra course but the author's aim is more ambitious - the book attempts to present both results and methods of abstract algebra as an exciting and beautiful part of modern mathematics. The fact that this edition is reprinted by the American Mathematical Society from the original that appeared in 1994, which became one of the most popular textbooks on algebra, reflects the fact that the author's aim has met with obvious success. The book is divided into two parts. The first of them is devoted to noncommutative algebra (following the author's professional interests and his conception of organisation of algebraic topics). The first ten chapters, almost the whole first third of the book, build group theory. The rest of the noncommutative part contains an introduction to the theory of modules and non-commutative rings. Elements of character theory, which forms a natural link between module theory and the theory of groups, are presented in the last chapter of the first part of the book. The commutative algebra part opens with a chapter devoted to polynomial rings, principal ideal rings and unique factorisation domains and is followed by a nine-chapter section covering field theory. The last five chapters present classical topics from commutative algebra and algebraic geometry, including primary decompositions of commutative noetherian rings, Dedekind domains and the Nullstellensatz.
The book covers almost all standard algebraic topics (except homological and categorical algebra) and, moreover, it includes several advanced parts of group theory (e.g. transfer theory). Note that the textbook is almost self-contained since a reader only needs an elementary algebraic background (elements of linear algebra and the basic concept of mathematical structures). Last but not least, a feature of the book that should be mentioned is the number of carefully chosen problems, which are listed at the end of every chapter. The textbook is excellent and an unusually successful combination of the best of traditional introduction to algebra, for which accuracy and correctness are absolutely essential, and a modern text that can be recommended to graduate students of algebra and all those whose interests lie in pure mathematics as well as everyone who wishes or needs to become familiar with the beauty of modern algebra.