×

Uh-oh, it looks like your Internet Explorer is out of date.

For a better shopping experience, please upgrade now.

Noncommutative Noetherian Rings
     

Noncommutative Noetherian Rings

by J. C. McConnell, J. C. Robson
 

ISBN-10: 0821821695

ISBN-13: 9780821821695

Pub. Date: 02/27/2001

Publisher: American Mathematical Society

Provides a comprehensive account of the major developments in this important branch of ring theory which have taken place over the past 30 years. Much of the material which comprises this volume has not appeared anywhere in book form before, and the authors have improved and simplified many of the accounts available in journals. The first few chapters form a ``basic

Overview

Provides a comprehensive account of the major developments in this important branch of ring theory which have taken place over the past 30 years. Much of the material which comprises this volume has not appeared anywhere in book form before, and the authors have improved and simplified many of the accounts available in journals. The first few chapters form a ``basic course'' which introduces the reader to the subject. Subsequent chapters are each relatively self-contained, so that readers interested in a particular subject can easily consult the sections they want. Specific topics covered include rings arising from matrices, differential operators, and Lie algebras. Contains extensive references.

Product Details

ISBN-13:
9780821821695
Publisher:
American Mathematical Society
Publication date:
02/27/2001
Series:
Graduate Studies in Mathematics Series , #30
Edition description:
New Edition
Pages:
636
Product dimensions:
7.09(w) x 10.63(h) x (d)

Table of Contents

Preface to the Revised Editionxiii
Prefacexv
Notationxix
Chapter 0.Preliminaries1
1.Chain Conditions1
2.Prime Radical4
3.Jacobson Radical6
Part I.Basic Theory
Chapter 1.Some Noetherian Rings11
1.Matrices11
2.Skew Polynomial Rings15
3.Weyl Algebras19
4.Skew Power Series and Laurent Polynomials21
5.Group Rings and Generalizations22
6.Skew Polynomial Rings in Several Variables26
7.Enveloping Algebras and Their Generalizations30
8.Further Simple Rings35
9.Additional Remarks38
Chapter 2.Quotient Rings and Goldie's Theorem41
1.Right Quotient Rings41
2.Uniform Dimension51
3.Goldie's Theorem57
4.Additional Remarks62
Chapter 3.Structure of Semiprime Goldie Rings63
1.Orders in Quotient Rings63
2.Minimal Primes70
3.Right Ideals in Semiprime Rings74
4.Endomorphism Rings78
5.Morita Equivalence83
6.Morita Contexts and Prime Rings89
7.Additional Remarks94
Chapter 4.Semiprime Ideals in Noetherian Rings97
1.Reduced Rank and Applications98
2.Artin-Rees Property and Localization106
3.Localization and Prime Ideals112
4.Affiliated Primes and Regular Elements123
5.Additivity128
6.Patch Continuity131
7.Additional Remarks138
Chapter 5.Some Dedekind-like Rings141
1.Maximal Orders142
2.Asano and Dedekind Prime Rings146
3.Classical Orders152
4.Hereditary Noetherian Rings159
5.Idealizer Rings162
6.Hereditary Noetherian Prime Rings166
7.Modules over Dedekind Prime Rings172
8.Additional Remarks180
Part II.Dimensions
Chapter 6.Krull Dimension185
1.Deviation of a Poset186
2.Krull Dimension of Modules192
3.Krull Dimension in Rings200
4.Prime Ideals and FBN Rings204
5.Bounds on Krull Dimensions209
6.Calculation of Krull Dimension213
7.Stable Bounds on Generators219
8.Quotient Rings and Localization223
9.Skew Polynomials over Commutative Rings233
10.Additional Remarks242
Chapter 7.Global Dimension245
1.Preliminaries245
2.Change of Rings250
3.Factor Rings253
4.Localization260
5.Estimates of Global Dimension261
6.Filtered and Graded Modules270
7.Regular Rings275
8.Fixed Rings278
9.Skew Polynomial Rings282
10.Skew Polynomials over Commutative Rings290
11.Simple Dedekind Domains291
12.Additional Remarks294
Chapter 8.Gelfand-Kirillov Dimension297
1.Definition and Examples298
2.Dimensions of Related Algebras304
3.Module Theory311
4.Almost Commutative Algebras and Hilbert Polynomials320
5.Applications324
6.Somewhat Commutative Algebras330
7.Additional Remarks337
Part III.Extensions
Chapter 9.The Nullstellensatz341
1.Algebras over a Field341
2.Algebras over Rings346
3.Generic Flatness348
4.Constructible Algebras351
5.Finite Dimensional Endomorphism Rings356
6.Polynomials over Division Rings358
7.Additional Remarks362
Chapter 10.Prime Ideals in Extension Rings365
1.Finite Extensions and Chain Conditions366
2.Prime Ideals370
3.Quotient Rings and Closure375
4.Incomparability380
5.Crossed Products and Fixed Rings387
6.Primes in Polynomial Rings392
7.Additional Remarks400
Chapter 11.Stability403
1.Stably Free Modules404
2.Stably Free Nonfree Modules409
3.Stable and Elementary Ranks413
4.Cancellation of Modules418
5.Ranks of Certain Rings421
6.Local Information428
7.Stability and Cancellation435
8.Additional Remarks443
Chapter 12.K[subscript 0] and Extension Rings445
1.K[subscript 0] of a Ring446
2.Projective-Graded Modules449
3.Filtrations and the Syzygy Theorem453
4.K[subscript 0] of Module Categories456
5.Skew Laurent Extensions461
6.Filtered Rings463
7.Applications to Simple Rings467
8.Additional Remarks472
Part IV.Examples
Chapter 13.Polynomial Identity Rings477
1.Polynomial Identities478
2.Nilpotence484
3.Central Simple Algebras486
4.Embeddings and Matrix Rings489
5.Central Polynomials492
6.Semiprime Rings and Central Polynomials497
7.Prime Ideals and Azumaya Algebras501
8.Integral Extensions and Prime Rings507
9.The Trace Ring and Maximal Orders514
10.Affine [kappa]-algebras518
11.Additional Remarks523
Chapter 14.Enveloping Algebras of Lie Algebras525
1.Basics526
2.Prime Ideals530
3.Eigenvalues and Prime Ideals534
4.Primitive Ideals537
5.The Solvable Case542
6.When Eigenvectors are Central545
7.When g is Algebraic550
8.The Simple Algebras A(V, [delta], [Gamma])552
9.The General Case557
10.Additional Remarks563
Chapter 15.Rings of Differential Operators on Algebraic Varieties565
1.Algebras over a Ring566
2.Affine Algebras over a Field573
3.Dimensions578
4.Further Properties585
5.Rings of Differential Operators590
6.Additional Remarks593
References595
Index of Notation621
Index625

Customer Reviews

Average Review:

Post to your social network

     

Most Helpful Customer Reviews

See all customer reviews