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Short description/annotation
This introduction to noncommutative noetherian rings is intended to be accessible to anyone with a basic background in abstract algebra.
Main description
This introduction to noncommutative noetherian rings is intended to be accessible to anyone with a basic background in abstract algebra. It can be used as a second-year graduate text, or as a self-contained reference. Extensive explanatory discussion is given, and exercises are integrated throughout. Various important settings, such as group algebras, Lie algebras, and quantum groups, are sketched at the outset to describe typical problems and provide motivation. The text then develops and illustrates the standard ingredients of the theory: e.g., skew polynomial rings, rings of fractions, bimodules, Krull dimension, linked prime ideals. Recurring emphasis is placed on prime ideals, which play a central role in applications to representation theory. This edition incorporates substantial revisions, particularly in the first third of the book, where the presentation has been changed to increase accessibility and topicality. New material includes the basic types of quantum groups, which then serve as test cases for the theory developed.
Table of contents:
1. A few Noetherian rings; 2. Skew polynomial rings; 3. Prime ideals; 4. Semisimple modules, Artinian modules, and torsionfree modules; 5. Injective hulls; 6. Semisimple rings of fractions; 7. Modules over semiprime Goldie rings; 8. Bimodules and affiliated prime ideals; 9. Fully bounded rings; 10. Rings and modules of fractions; 11. Artinian quotient rings; 12. Links between prime ideals; 13. The Artin-Rees property; 14. Rings satisfying the second layer condition; 15. Krull dimension; 16. Numbers of generators of modules; 17. Transcendental division algebras.
This introduction to noncommutative noetherian rings is intended to be accessible to anyone with a basic background in abstract algebra.
Main description
This introduction to noncommutative noetherian rings is intended to be accessible to anyone with a basic background in abstract algebra. It can be used as a second-year graduate text, or as a self-contained reference. Extensive explanatory discussion is given, and exercises are integrated throughout. Various important settings, such as group algebras, Lie algebras, and quantum groups, are sketched at the outset to describe typical problems and provide motivation. The text then develops and illustrates the standard ingredients of the theory: e.g., skew polynomial rings, rings of fractions, bimodules, Krull dimension, linked prime ideals. Recurring emphasis is placed on prime ideals, which play a central role in applications to representation theory. This edition incorporates substantial revisions, particularly in the first third of the book, where the presentation has been changed to increase accessibility and topicality. New material includes the basic types of quantum groups, which then serve as test cases for the theory developed.
Table of contents:
1. A few Noetherian rings; 2. Skew polynomial rings; 3. Prime ideals; 4. Semisimple modules, Artinian modules, and torsionfree modules; 5. Injective hulls; 6. Semisimple rings of fractions; 7. Modules over semiprime Goldie rings; 8. Bimodules and affiliated prime ideals; 9. Fully bounded rings; 10. Rings and modules of fractions; 11. Artinian quotient rings; 12. Links between prime ideals; 13. The Artin-Rees property; 14. Rings satisfying the second layer condition; 15. Krull dimension; 16. Numbers of generators of modules; 17. Transcendental division algebras.