Noncommutative Noetherian Rings

By: John C. McConnell, J.C. Robson

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Hardcover | 1 January 2001

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This is a reprinted edition of a work that was considered the definitive account in the subject area upon its initial publication by J. Wiley & Sons in 1987. It presents, within a wider context, a comprehensive account of noncommutative Noetherian rings. The author covers the major developments from the 1950s, stemming from Goldie's theorem and onward, including applications to group rings, enveloping algebras of Lie algebras, PI rings, differential operators, and localization theory. The book is not restricted to Noetherian rings, but discusses wider classes of rings where the methods apply more generally. In the current edition, some errors were corrected, a number of arguments have been expanded, and the references were brought up to date. This reprinted edition will continue to be a valuable and stimulating work for readers interested in ring theory and its applications to other areas of mathematics.

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Non-Fiction Mathematics Algebra

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Non-Fiction Mathematics Algebra

Preface to the Revised Edition	p. xiii
Preface	p. xv
Notation	p. xix
Preliminaries	p. 1
Chain Conditions	p. 1
Prime Radical	p. 4
Jacobson Radical	p. 6
Basic Theory
Some Noetherian Rings	p. 11
Matrices	p. 11
Skew Polynomial Rings	p. 15
Weyl Algebras	p. 19
Skew Power Series and Laurent Polynomials	p. 21
Group Rings and Generalizations	p. 22
Skew Polynomial Rings in Several Variables	p. 26
Enveloping Algebras and Their Generalizations	p. 30
Further Simple Rings	p. 35
Additional Remarks	p. 38
Quotient Rings and Goldie's Theorem	p. 41
Right Quotient Rings	p. 41
Uniform Dimension	p. 51
Goldie's Theorem	p. 57
Additional Remarks	p. 62
Structure of Semiprime Goldie Rings	p. 63
Orders in Quotient Rings	p. 63
Minimal Primes	p. 70
Right Ideals in Semiprime Rings	p. 74
Endomorphism Rings	p. 78
Morita Equivalence	p. 83
Morita Contexts and Prime Rings	p. 89
Additional Remarks	p. 94
Semiprime Ideals in Noetherian Rings	p. 97
Reduced Rank and Applications	p. 98
Artin-Rees Property and Localization	p. 106
Localization and Prime Ideals	p. 112
Affiliated Primes and Regular Elements	p. 123
Additivity	p. 128
Patch Continuity	p. 131
Additional Remarks	p. 138
Some Dedekind-like Rings	p. 141
Maximal Orders	p. 142
Asano and Dedekind Prime Rings	p. 146
Classical Orders	p. 152
Hereditary Noetherian Rings	p. 159
Idealizer Rings	p. 162
Hereditary Noetherian Prime Rings	p. 166
Modules over Dedekind Prime Rings	p. 172
Additional Remarks	p. 180
Dimensions
Krull Dimension	p. 185
Deviation of a Poset	p. 186
Krull Dimension of Modules	p. 192
Krull Dimension in Rings	p. 200
Prime Ideals and FBN Rings	p. 204
Bounds on Krull Dimensions	p. 209
Calculation of Krull Dimension	p. 213
Stable Bounds on Generators	p. 219
Quotient Rings and Localization	p. 223
Skew Polynomials over Commutative Rings	p. 233
Additional Remarks	p. 242
Global Dimension	p. 245
Preliminaries	p. 245
Change of Rings	p. 250
Factor Rings	p. 253
Localization	p. 260
Estimates of Global Dimension	p. 261
Filtered and Graded Modules	p. 270
Regular Rings	p. 275
Fixed Rings	p. 278
Skew Polynomial Rings	p. 282
Skew Polynomials over Commutative Rings	p. 290
Simple Dedekind Domains	p. 291
Additional Remarks	p. 294
Gelfand-Kirillov Dimension	p. 297
Definition and Examples	p. 298
Dimensions of Related Algebras	p. 304
Module Theory	p. 311
Almost Commutative Algebras and Hilbert Polynomials	p. 320
Applications	p. 324
Somewhat Commutative Algebras	p. 330
Additional Remarks	p. 337
Extensions
The Nullstellensatz	p. 341
Algebras over a Field	p. 341
Algebras over Rings	p. 346
Generic Flatness	p. 348
Constructible Algebras	p. 351
Finite Dimensional Endomorphism Rings	p. 356
Polynomials over Division Rings	p. 358
Additional Remarks	p. 362
Prime Ideals in Extension Rings	p. 365
Finite Extensions and Chain Conditions	p. 366
Prime Ideals	p. 370
Quotient Rings and Closure	p. 375
Incomparability	p. 380
Crossed Products and Fixed Rings	p. 387
Primes in Polynomial Rings	p. 392
Additional Remarks	p. 400
Stability	p. 403
Stably Free Modules	p. 404
Stably Free Nonfree Modules	p. 409
Stable and Elementary Ranks	p. 413
Cancellation of Modules	p. 418
Ranks of Certain Rings	p. 421
Local Information	p. 428
Stability and Cancellation	p. 435
Additional Remarks	p. 443
K[subscript 0] and Extension Rings	p. 445
K[subscript 0] of a Ring	p. 446
Projective-Graded Modules	p. 449
Filtrations and the Syzygy Theorem	p. 453
K[subscript 0] of Module Categories	p. 456
Skew Laurent Extensions	p. 461
Filtered Rings	p. 463
Applications to Simple Rings	p. 467
Additional Remarks	p. 472
Examples
Polynomial Identity Rings	p. 477
Polynomial Identities	p. 478
Nilpotence	p. 484
Central Simple Algebras	p. 486
Embeddings and Matrix Rings	p. 489
Central Polynomials	p. 492
Semiprime Rings and Central Polynomials	p. 497
Prime Ideals and Azumaya Algebras	p. 501
Integral Extensions and Prime Rings	p. 507
The Trace Ring and Maximal Orders	p. 514
Affine [kappa]-algebras	p. 518
Additional Remarks	p. 523
Enveloping Algebras of Lie Algebras	p. 525
Basics	p. 526
Prime Ideals	p. 530
Eigenvalues and Prime Ideals	p. 534
Primitive Ideals	p. 537
The Solvable Case	p. 542
When Eigenvectors are Central	p. 545
When g is Algebraic	p. 550
The Simple Algebras A(V, [delta], [Gamma])	p. 552
The General Case	p. 557
Additional Remarks	p. 563
Rings of Differential Operators on Algebraic Varieties	p. 565
Algebras over a Ring	p. 566
Affine Algebras over a Field	p. 573
Dimensions	p. 578
Further Properties	p. 585
Rings of Differential Operators	p. 590
Additional Remarks	p. 593
References	p. 595
Index of Notation	p. 621
Index	p. 625
Table of Contents provided by Syndetics. All Rights Reserved.

Noncommutative Noetherian Rings

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