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Book DetailsGraduate Studies in MathematicsVolume: 91; 2008; 648 ppMSC: Primary 16; 17; Secondary 20
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 onesemester 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.
ReadershipGraduate students and research mathematicians interested in various topics of noncommutative algebra.

Table of 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 WedderburnArtin 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 (CoxeterDynkin 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


Additional Material

Reviews

Each part ends with more than 30 pages of exercises, from the basic to the challenging, carefully arranged and labeled according to the chapter (or appendix) to which they related ... a striking and very enjoyable feature of the book is the huge number of digressions: there are frequent pauses to point out noteworthy aspects of the terrain which lies ahead, beyond what can be covered in detail in a book of this sort. The style, layout and precision of the book make it a pleasure to read...
Mathematical Reviews 
The book is largely selfcontained. ...a valuable textbook and a reliable reference for graduate students.
MAA Reviews


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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 onesemester 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.
Graduate students and research mathematicians interested in various topics of noncommutative algebra.

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 WedderburnArtin 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 (CoxeterDynkin 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

Each part ends with more than 30 pages of exercises, from the basic to the challenging, carefully arranged and labeled according to the chapter (or appendix) to which they related ... a striking and very enjoyable feature of the book is the huge number of digressions: there are frequent pauses to point out noteworthy aspects of the terrain which lies ahead, beyond what can be covered in detail in a book of this sort. The style, layout and precision of the book make it a pleasure to read...
Mathematical Reviews 
The book is largely selfcontained. ...a valuable textbook and a reliable reference for graduate students.
MAA Reviews