This book presents, within a wider context, a comprehensive account of noncommutative Noetherian rings. It covers the major developments of the last 30 years which stem from Goldie's theorem, 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 when the methods being described apply more generally. Most of the material in the book has not previously appeared outside the research literature, and considerable revisions, improvements and extensions have been made. The later chapters are designed to be more or less self-contained, so that readers can easily consult the book on particular topics. There are notes linking each chapter to the research literature and a substantial list of references.
BASIC THEORY: Some Noetherian Rings; Quotient Rings and Goldie's Theorem; Structure of Semiprime Goldie Rings; Semiprime Ideals in Noetherian Rings; Some Dedekind-Like Rings; DIMENSIONS: Krull Dimension; Global Dimension; Gelfand Kirillov Dimension; EXTENSIONS: The Nullstellensatz; Prime Ideals in Extension Rings; Stability; K0 and Extension Rings; EXAMPLES: Polynomial Identity Rings; Enveloping Algebras of Lie Algebras; Rings of Differential Operators on Algebraic Varieties; References; Index of Notation; Index.