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This work specifically surveys simple Noetherian rings. The authors present theorems on the structure of simple right Noetherian rings and, more generally, on simple rings containing a uniform right ideal U. The text is as elementary and self-contained as practicable, and the little background required in homological and categorical algebra is given in a short appendix. Full definitions are given and short, complete, elementary proofs are provided for such key theorems as the Morita theorem, the Correspondence theorem, the Wedderburn-Artin theorem, the Goldie-Lesieur-Croisot theorem, and many others. Complex mathematical machinery has been eliminated wherever possible or its introduction into the text delayed as long as possible. (Even tensor products are not required until Chapter 3.)

Noetherian rings.

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Commutative rings --- Noetherian rings --- Krull rings

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Ordered algebraic structures --- Noetherian rings. --- Rings (Algebra) --- Anneaux noethériens --- Anneaux (algèbre) --- Noetherian rings --- Representations of rings (Algebra) --- Rings, Noetherian --- Associative rings --- Commutative rings --- Anneaux noethériens.

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This 2004 introduction to noncommutative noetherian rings is intended to be accessible to anyone with a basic background in abstract algebra. It can be used as a second-year graduate text, or as a self-contained reference. Extensive explanatory discussion is given, and exercises are integrated throughout. Various important settings, such as group algebras, Lie algebras, and quantum groups, are sketched at the outset to describe typical problems and provide motivation. The text then develops and illustrates the standard ingredients of the theory: e.g., skew polynomial rings, rings of fractions, bimodules, Krull dimension, linked prime ideals. Recurring emphasis is placed on prime ideals, which play a central role in applications to representation theory. This edition incorporates substantial revisions, particularly in the first third of the book, where the presentation has been changed to increase accessibility and topicality. Material includes the basic types of quantum groups, which then serve as test cases for the theory developed.

Noetherian rings. --- Noncommutative rings. --- Non-commutative rings --- Associative rings --- Rings, Noetherian --- Commutative rings

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Anneaux noethériens --- Noetherian rings. --- Algèbres commutatives --- Corps algébriques --- Ideaux et modules