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Author Goodearl, K. R.

Title An introduction to noncommutative Noetherian rings / K.R. Goodearl, R.B. Warfield, Jr.

Publication Info. Cambridge, U.K. ; New York : Cambridge University Press, 2004.

Item Status

Edition 2nd ed.
Description 1 online resource (xxiv, 344 pages).
data file
Physical Medium polychrome
Series London Mathematical Society student texts ; 61
London Mathematical Society student texts ; 61.
Bibliography Includes bibliographical references (pages 328-337) and index.
Summary 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.
Contents Cover -- Series-title -- Title -- Copyright -- Contents -- Introduction to the Second Edition -- Introduction to the First Edition -- Prologue -- POLYNOMIAL IDENTITY RINGS -- GROUP ALGEBRAS -- RINGS OF DIFFERENTIAL OPERATORS -- ENVELOPING ALGEBRAS -- QUANTUM GROUPS -- NOTATION AND TERMINOLOGY -- 1. A Few Noetherian Rings -- THE NOETHERIAN CONDITION -- FORMAL TRIANGULAR MATRIX RINGS -- THE HILBERT BASIS THEOREM -- SKEW POLYNOMIAL RINGS TWISTED BY AUTOMORPHISMS -- SKEW-LAURENT RINGS -- SIMPLICITY IN SKEW-LAURENT RINGS -- ADDITIONAL EXERCISES.
NOTES -- 2. Skew Polynomial Rings -- FORMAL DIFFERENTIAL OPERATOR RINGS -- WEYL ALGEBRAS -- GENERAL SKEW POLYNOMIAL RINGS -- A GENERAL SKEW HILBERT BASIS THEOREM -- SOME EXAMPLES -- ADDITIONAL EXERCISES -- NOTES -- 3. Prime Ideals -- PRIME IDEALS -- SEMIPRIME IDEALS AND NILPOTENCE -- ANNIHILATORS AND ASSOCIATED PRIME IDEALS -- SOME EXAMPLES FROM REPRESENTATION THEORY -- PRIMITIVE AND SEMIPRIMITIVE IDEALS -- PRIME IDEALS IN DIFFERENTIAL OPERATOR RINGS -- ADDITIONAL EXERCISES -- NOTES.
4. Semisimple Modules, Artinian Modules, and Torsionfree Modules -- SEMISIMPLE MODULES -- SEMISIMPLE RINGS -- ARTINIAN MODULES -- ARTINIAN RINGS -- TORSION AND TORSIONFREE MODULES -- NOTES -- 5. Injective Hulls -- INJECTIVE MODULES -- ESSENTIAL EXTENSIONS -- INJECTIVE HULLS -- MODULES OF FINITE RANK -- UNIFORM RANK -- DIRECT SUMS OF INJECTIVE MODULES -- ASSASSINATOR PRIMES -- ADDITIONAL EXERCISES -- NOTES -- 6. Semisimple Rings of Fractions -- RINGS OF FRACTIONS -- DIVISION RINGS OF FRACTIONS -- GOLDIE'S THEOREM.
NIL SUBSETS -- ADDITIONAL EXERCISES -- NOTES -- 7. Modules over Semiprime Goldie Rings -- MINIMAL PRIME IDEALS -- TORSION -- TORSIONFREE INJECTIVE MODULES -- TORSIONFREE UNIFORM MODULES -- TORSIONFREE MODULES OVER PRIME GOLDIE RINGS -- NOTES -- 8. Bimodules and A.liated Prime Ideals -- NOETHERIAN BIMODULES -- AFFILIATED PRIME IDEALS -- ARTINIAN BIMODULES -- PRIME IDEALS IN FINITE RING EXTENSIONS -- BIMODULE COMPOSITION SERIES -- ADDITIVITY PRINCIPLES -- NORMALIZING EXTENSIONS -- NOTES -- 9. Fully Bounded Rings -- BOUNDEDNESS.
EMBEDDING MODULES INTO FACTOR RINGS -- ARTINIAN MODULES -- UNIFORM INJECTIVE MODULES -- NOTES -- 10. Rings and Modules of Fractions -- RINGS OF FRACTIONS -- MODULES OF FRACTIONS -- SUBMODULES OF MODULES OF FRACTIONS -- IDEALS IN RINGS OF FRACTIONS -- PRIME IDEALS IN ITERATED DIFFERENTIAL OPERATOR RINGS -- ADDITIONAL EXERCISES -- NOTES -- 11. Artinian Quotient Rings -- REDUCED RANK -- APPLICATIONS OF REDUCED RANK TO FINITE RING EXTENSIONS -- SMALL'S THEOREM -- AFFILIATED PRIME IDEALS -- AFFILIATED PRIME IDEALS -- NOTES.
Local Note eBooks on EBSCOhost EBSCO eBook Subscription Academic Collection - North America
Subject Noetherian rings.
Noetherian rings.
Noncommutative rings.
Noncommutative rings.
Genre/Form Electronic books.
Electronic books.
Added Author Warfield, Robert B., 1940-
Other Form: Print version: Goodearl, K.R. Introduction to noncommutative Noetherian rings. 2nd ed. Cambridge, U.K. ; New York : Cambridge University Press, 2004 0521836875 (DLC) 2003065357 (OCoLC)53231332
ISBN 0511211929
9780511211928
0521836875
9780521836876
0521545374 (paperback)
9780521545372 (paperback)
0511217293 (electronic book)
9780511217296 (electronic book)
0511215509 (electronic book)
9780511215506 (electronic book)
9780511841699
0511841698