Description
This is an updated 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.
Chapter
Preface to the Revised Edition
Chapter 1. Some Noetherian Rings
§2. Skew Polynomial Rings
§4. Skew Power Series and Laurent Polynomials
§5. Group Rings and Generalizations
§6. Skew Polynomial Rings in Several Variables
§7. Enveloping Algebras and Their Generalizations
Chapter 2. Quotient Rings and Goldie's Theorem
Chapter 3. Structure of Semiprime Goldie Rings
§1. Orders in Quotient Rings
§3. Right Ideals in Semiprime Rings
§6. Morita Contexts and Prime Rings
Chapter 4. Semiprime Ideals in Noetherian Rings
§1. Reduced Rank and Applications
§2. Artin-Rees Property and Localization
§3. Localization and Prime Ideals
§4. Affiliated Primes and Regular Elements
Chapter 5. Some Dedekind-like Rings
§2. Asano and Dedekind Prime Rings
§4. Hereditary Noetherian Rings
§6. Hereditary Noetherian Prime Rings
§7. Modules over Dedekind Prime Rings
Chapter 6. Krull Dimension
§2. Krull Dimension of Modules
§3. Krull Dimension in Rings
§4. Prime Ideals and FBN Rings
§5. Bounds on Krull Dimensions
§6. Calculation of Krull Dimension
§7. Stable Bounds on Generators
§8. Quotient Rings and Localization
§9. Skew Polynomials over Commutative Rings
Chapter 7. Global Dimension
§5. Estimates of Global Dimension
§6. Filtered and Graded Modules
§9. Skew Polynomial Rings
§10. Skew Polynomials over Commutative Rings
§11. Simple Dedekind Domains
Chapter 8. Gelfand-Kirillov Dimension
§1. Definition and Examples
§2. Dimensions of Related Algebras
§4. Almost Commutative Algebras and Hilbert Polynomials
§6. Somewhat Commutative Algebras
Chapter 9. The Nullstellensatz
§1. Algebras over a Field
§4. Constructible Algebras
§5. Finite Dimensional Endomorphism Rings
§6. Polynomials over Division Rings
Chapter 10. Prime Ideals in Extension Rings
§1. Finite Extensions and Chain Conditions
§3. Quotient Rings and Closure
§5. Crossed Products and Fixed Rings
§6. Primes in Polynomial Rings
§2. Stably Free Nonfree Modules
§3. Stable and Elementary Ranks
§4. Cancellation of Modules
§5. Ranks of Certain Rings
§7. Stability and Cancellation
Chapter 12. K[sub(0)] and Extension Rings
§2. Projective-Graded Modules
§3. Filtrations and the Syzygy Theorem
§4. K[sub(0)] of Module Categories
§5. Skew Laurent Extensions
§7. Applications to Simple Rings
Chapter 13. Polynomial Identity Rings
§1. Polynomial Identities
§3. Central Simple Algebras
§4. Embeddings and Matrix Rings
§6. Semiprime Rings and Central Polynomials
§7. Prime Ideals and Azumaya Algebras
§8. Integral Extensions and Prime Rings
§9. The Trace Ring and Maximal Orders
Chapter 14. Enveloping Algebras of Lie Algebras
§3. Eigenvalues and Prime Ideals
§6. When Eigenvectors are Central
§8. The Simple Algebras A(V,δ,Γ)
Chapter 15. Rings of Differential Operators on Algebraic Varieties
§2. Affine Algebras over a Field
§5. Rings of Differential Operators
Noncommutative Noetherian Rings
( Graduate Studies in Mathematics )
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