**Mathematical Surveys and Monographs**

Volume: 174;
2011;
228 pp;
Hardcover

MSC: Primary 16; 18; 19;

Print ISBN: 978-0-8218-5350-4

Product Code: SURV/174

List Price: $89.00

Individual Member Price: $71.20

**Electronic ISBN: 978-1-4704-1401-6
Product Code: SURV/174.E**

List Price: $89.00

Individual Member Price: $71.20

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#### Supplemental Materials

# Hereditary Noetherian Prime Rings and Idealizers

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*Lawrence S. Levy; J. Chris Robson*

The direct sum behaviour of its projective modules is a fundamental property of any ring. Hereditary Noetherian prime rings are perhaps the only noncommutative Noetherian rings for which this direct sum behaviour (for both finitely and infinitely generated projective modules) is well-understood, yet highly nontrivial.

This book surveys material previously available only in the research literature. It provides a re-worked and simplified account, with improved clarity, fresh insights and many original results about finite length modules, injective modules and projective modules. It culminates in the authors' surprisingly complete structure theorem for projective modules which involves two independent additive invariants: genus and Steinitz class. Several applications demonstrate its utility.

The theory, extending the well-known module theory of commutative Dedekind domains and of hereditary orders, develops via a detailed study of simple modules. This relies upon the substantial account of idealizer subrings which forms the first part of the book and provides a useful general construction tool for interesting examples.

The book assumes some knowledge of noncommutative Noetherian rings, including Goldie's theorem. Beyond that, it is largely self-contained, thanks to the appendix which provides succinct accounts of Artinian serial rings and, for arbitrary rings, results about lifting direct sum decompositions from finite length images of projective modules. The appendix also describes some open problems.

The history of the topics is surveyed at appropriate points.

#### Table of Contents

# Table of Contents

## Hereditary Noetherian Prime Rings and Idealizers

- Cover Cover11 free
- Title page i2 free
- Contents iii4 free
- Introduction and standard notation 16 free
- Part I. Idealizer rings 510 free
- Basic idealizers 712
- Iterated and multichain idealizers 2732
- Part II. HNP rings 4752
- Basic structure 4954
- Towers 7782
- Integral overrings 113118
- Invariants for finitely generated projective modules 131136
- Applications of invariants 153158
- Infinitely generated projective modules 175180
- Related topics 193198
- Bibliography 219224
- Index of symbols 223228 free
- Index of terminology 225230 free
- Back Cover Back Cover1234

#### Readership

Graduate students and research mathematicians interested in algebra, in particular, noncommutative rings.