Hereditary noetherian prime rings and idealizers / Lawrence S. Levy, J. Chris Robson.

Format:

Book

Author/Creator:

Levy, Lawrence S., 1933- author.

Robson, J. C. (James Christopher), 1940- author.

Series:

Mathematical surveys and monographs ; Volume 174.

Mathematical surveys and monographs ; Volume 174

Language:

English

Subjects (All):

Algebra, Homological.

Grothendieck groups.

Physical Description:

1 online resource (iv, 228 pages) : illustrations.

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2011]

Language Note:

English

Summary:

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.

Contents:

Cover

Title page

Contents

Introduction and standard notation

Part I. Idealizer rings

Basic idealizers

Iterated and multichain idealizers

Part II. HNP rings

Basic structure

Towers

Integral overrings

Invariants for finitely generated projective modules

Applications of invariants

Infinitely generated projective modules

Related topics

Bibliography

Index of symbols

Index of terminology

Back Cover.

Notes:

Bibliographic Level Mode of Issuance: Monograph

Includes bibliographical references (pages 219-221) and indexes.

Description based on print version record.

Description based on publisher supplied metadata and other sources.

ISBN:

1-4704-1401-5

OCLC:

898200126