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. Readership Graduate students and research mathematicians interested in ring theory and its applications. Reviews "A comprehensive list of references (containing more than 500 items) is included."  Mathematica Bohemica From reviews of the first edition ... "A model of mathematical writing, as perfectly written a mathematics book as I have seen ... It can be profitably read by nonexperts ... an almost perfectly conceived account of major developments and general methods ... will remain a basic reference for many years ..."  Bulletin of the AMS "Very thorough and illuminating ... A veritable tour de force, encompassing a wide range of topics in some depth ... very easy to find information in this book ... full of illuminating examples which throw a light on [the theory]."  Proceedings of the Edinburgh Mathematical Society "Selfcontained, comprehensive ... The creation of this valuable resource is a service to mathematics ..."  Mathematical Reviews "An intrinsically interesting branch of algebra ... Until ... this book there has been no attempt to provide an overview of, and a general reference for, the most important developments in the theory. The ... authors set out to fill this gap and have succeeded admirably ... easy to read and use ... well written ... An essential possession for any serious worker in the area."  Zentralblatt MATH "An account of noncommutative Noetherian rings, giving the theory as far as it exists but with constant emphasis on constructions and examples. [This is] a daunting task but the authors have succeeded well ... highly readable ... well indexed ... will rapidly become the standard text in the field and will stimulate further progress."  Bulletin of the LMS "An abundance of wellorganized material ... a must for those in the area."  International Mathematical News Table of Contents Basic theory  Some Noetherian rings
 Quotient rings and Goldie's theorem
 Structure of semiprime Goldie rings
 Semiprime ideals in Noetherian rings
 Some Dedekindlike rings
Dimensions  Krull dimension
 Global dimension
 GelfandKirillov dimension
Extensions  The Nullstellensatz
 Prime ideals in extension rings
 Stability
 \(K_0\) and extension rings
Examples  Polynomial identity rings
 Enveloping algebras of Lie algebras
 Rings of differential operators on algebraic varieties
 References
 Index of notation
 Index
