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Hereditary Noetherian Prime Rings and Idealizers

Lawrence S. Levy University of Wisconsin, Madison, WI
J. Chris Robson University of Leeds, Leeds, United Kingdom
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Hardcover ISBN: 978-0-8218-5350-4
Product Code: SURV/174
List Price: $95.00 MAA Member Price:$85.50
AMS Member Price: $76.00 Electronic ISBN: 978-1-4704-1401-6 Product Code: SURV/174.E List Price:$89.00
MAA Member Price: $80.10 AMS Member Price:$71.20
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List Price: $142.50 MAA Member Price:$128.25
AMS Member Price: $114.00 Click above image for expanded view Hereditary Noetherian Prime Rings and Idealizers Lawrence S. Levy University of Wisconsin, Madison, WI J. Chris Robson University of Leeds, Leeds, United Kingdom Available Formats:  Hardcover ISBN: 978-0-8218-5350-4 Product Code: SURV/174  List Price:$95.00 MAA Member Price: $85.50 AMS Member Price:$76.00
 Electronic ISBN: 978-1-4704-1401-6 Product Code: SURV/174.E
 List Price: $89.00 MAA Member Price:$80.10 AMS Member Price: $71.20 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$142.50 MAA Member Price: $128.25 AMS Member Price:$114.00
• Book Details

Mathematical Surveys and Monographs
Volume: 1742011; 228 pp
MSC: 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 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.

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

• Requests

Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 1742011; 228 pp
MSC: 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 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.

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
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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