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Full Description
This book is a companion volume to Graduate Algebra: Commutative View (published as volume 73 in this series). The main and most important feature of the book is that it presents a unified approach to many important topics, such as group theory, ring theory, Lie algebras, and gives conceptual proofs of many basic results of noncommutative algebra. There are also a number of major results in noncommutative algebra that are usually found only in technical works, such as Zelmanov's proof of the restricted Burnside problem in group theory, word problems in groups, Tits's alternative in algebraic groups, PI algebras, and many of the roles that Coxeter diagrams play in algebra. The first half of the book can serve as a one-semester course on noncommutative algebra, whereas the remaining part of the book describes some of the major directions of research in the past 100 years. The main text is extended through several appendices, which permits the inclusion of more advanced material, and numerous exercises. The only prerequisite for using the book is an undergraduate course in algebra; whenever necessary, results are quoted from Graduate Algebra: Commutative View.
Contents
Part IV. The structure of rings
Introduction to the structure of rings
Chapter 13. Fundamental concepts in ring theory
Chapter 14. Semisimple modules and rings and the Wedderburn-Artin theorem
Chapter 15. The Jacobson program applied to left Artinian rings
Chapter 16. Noetherian rings and the role of prime rings
Chapter 17. Algebras in terms of generators and relations
Chapter 18. Tensor products
Exercises-Part IV
Part V. Representations of groups and Lie algebras
Introduction to representations of groups and Lie algebras
Chapter 19. Group representations and group algebras
Chapter 20. Characters of finite groups
Chapter 21. Lie algebras and other nonassociative algebras
Chapter 22. Dynkin diagrams (Coxeter-Dynkin graphs and Coxeter groups)
Exercises-Part V
Part VI. Representable algebras
Introduction to representable algebras
Chapter 23. Polynomial identities and representable algebras
Chapter 24. Central simple algebras and the Brauer group
Chapter 25. Homological algebra and categories of modules
Chapter 26. Hopf algebras
Exercises-Part VI
