Description
This 2004 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. Material includes the basic types of quantum groups, which then serve as test cases for the theory developed.
Chapter
• SIMPLICITY IN SKEW-LAURENT RINGS •
• FORMAL DIFFERENTIAL OPERATOR RINGS •
• GENERAL SKEW POLYNOMIAL RINGS •
• A GENERAL SKEW HILBERT BASIS THEOREM •
• SEMIPRIME IDEALS AND NILPOTENCE •
• ANNIHILATORS AND ASSOCIATED PRIME IDEALS •
• SOME EXAMPLES FROM REPRESENTATION THEORY •
• PRIMITIVE AND SEMIPRIMITIVE IDEALS •
• PRIME IDEALS IN DIFFERENTIAL OPERATOR RINGS •
4. Semisimple Modules, Artinian Modules, and Torsionfree Modules
• TORSION AND TORSIONFREE MODULES •
• MODULES OF FINITE RANK •
• DIRECT SUMS OF INJECTIVE MODULES •
6. Semisimple Rings of Fractions
• DIVISION RINGS OF FRACTIONS •
7. Modules over Semiprime Goldie Rings
• TORSIONFREE INJECTIVE MODULES •
• TORSIONFREE UNIFORM MODULES •
• TORSIONFREE MODULES OVER PRIME GOLDIE RINGS •
8. Bimodules and A.liated Prime Ideals
• AFFILIATED PRIME IDEALS •
• PRIME IDEALS IN FINITE RING EXTENSIONS •
• BIMODULE COMPOSITION SERIES •
• ADDITIVITY PRINCIPLES •
• NORMALIZING EXTENSIONS •
• EMBEDDING MODULES INTO FACTOR RINGS •
• UNIFORM INJECTIVE MODULES •
10. Rings and Modules of Fractions
• SUBMODULES OF MODULES OF FRACTIONS •
• IDEALS IN RINGS OF FRACTIONS •
• PRIME IDEALS IN ITERATED DIFFERENTIAL OPERATOR RINGS •
11. Artinian Quotient Rings
• APPLICATIONS OF REDUCED RANK TO FINITE RING EXTENSIONS •
• AFFILIATED PRIME IDEALS •
• AFFILIATED PRIME IDEALS •
12. Links Between Prime Ideals
• LINKS AND SHORT AFFILIATED SERIES •
• LINKS AND AFFILIATED PRIMES •
13. The Artin-Rees Property
• THE ARTIN-REES PROPERTY •
• MODULE-FINITE ALGEBRAS •
14. Rings Satisfying the Second Layer Condition
• CLASSICAL KRULL DIMENSION •
• BIMODULE SYMMETRY AND INTERSECTION THEOREMS •
• FINITE RING EXTENSIONS •
• LOCALIZATION AT A SEMIPRIME IDEAL •
• EMBEDDINGS INTO ARTINIAN RINGS •
• LOCALIZATION AT INFINITE SETS OF PRIME IDEALS •
• DEFINITIONS AND BASIC PROPERTIES •
• PRIME NOETHERIAN RINGS •
• CRITICAL COMPOSITION SERIES •
• POLYNOMIAL AND SKEW POLYNOMIAL RINGS •
16. Numbers of Generators of Modules
• TOPOLOGIES ON THE PRIME SPECTRUM •
• LOCAL NUMBERS OF GENERATORS •
• PATCH-CONTINUITY OF NORMALIZED RANKS •
• GENERATING MODULES OVER SIMPLE NOETHERIAN RINGS •
• GENERATING MODULES OVER FBN RINGS •
• COUNTABILITY OF CLIQUES •
17. Transcendental Division Algebras
• POLYNOMIALS OVER DIVISION RINGS •
• FULLY BOUNDED G-RINGS •
• PRIMITIVITY AND TRANSCENDENCE DEGREE •
• QUOTIENT DIVISION RINGS OF WEYL ALGEBRAS AND QUANTUM PLANES •
• FINITE GENERATION OF SUBFIELDS •
Appendix. Some Test Problems for Noetherian Rings
1. Jacobson’s Conjecture.
2. The Artin-Rees Property for Jacobson Radicals.
3. The Descending Chain Condition for Prime Ideals.
4. Countability of Chains of Ideals or Submodules.
5. Local Rings of Finite Global Dimension.
6. Krull Versus Global Dimension.
7. Global Dimension via Simple Modules.
8. Finite Projective Dimension for Finitely Generated Modules.
10. Transfer Across Noetherian Bimodules.
11. Incomparability in Cliques.
12. Localizability of Cliques.
13. Prime Middle Annihilators.
14. Nilpotence Modulo One-Sided Ideals.
15. Extension of a Base Field.
16. Tensor Products of Noetherian Algebras.
17. Classical Krull Dimension of Polynomial Rings.
18. Symmetry of Primitivity.
19. Generating Right Ideals in Simple Noetherian Rings.
20. Torsionfree Modules over Simple Noetherian Rings.
An Introduction to Noncommutative Noetherian Rings
( London Mathematical Society Student Texts )
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