Hardcover ISBN:  9780821853504 
Product Code:  SURV/174 
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Electronic ISBN:  9781470414016 
Product Code:  SURV/174.E 
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Book DetailsMathematical Surveys and MonographsVolume: 174; 2011; 228 ppMSC: Primary 16; 18; 19;
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 wellunderstood, yet highly nontrivial.
This book surveys material previously available only in the research literature. It provides a reworked 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 wellknown 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 selfcontained, 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.ReadershipGraduate students and research mathematicians interested in algebra, in particular, noncommutative rings.

Table of Contents

Chapters

Introduction and standard notation

Part 1. Idealizer rings

1. Basic idealizers

2. Iterated and multichain idealizers

Part 2. HNP rings

3. Basic structure

4. Towers

5. Integral overrings

6. Invariants for finitely generated projective modules

7. Applications of invariants

8. Infinitely generated projective modules

9. Related topics


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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 wellunderstood, yet highly nontrivial.
This book surveys material previously available only in the research literature. It provides a reworked 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 wellknown 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 selfcontained, 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.
Graduate students and research mathematicians interested in algebra, in particular, noncommutative rings.

Chapters

Introduction and standard notation

Part 1. Idealizer rings

1. Basic idealizers

2. Iterated and multichain idealizers

Part 2. HNP rings

3. Basic structure

4. Towers

5. Integral overrings

6. Invariants for finitely generated projective modules

7. Applications of invariants

8. Infinitely generated projective modules

9. Related topics