Hereditary Noetherian Prime Rings and Idealizers
( Mathematical Surveys and Monographs )
Publication series ：Mathematical Surveys and Monographs
Author： Lawrence S. Levy;J. Chris Robson
Publisher： American Mathematical Society
Publication year： 2011
E-ISBN: 9781470414016
P-ISBN(Paperback): 9780821853504
Subject： O154.2 homological algebra
Keyword： Algebra and Algebraic Geometry
Language： ENG
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Hereditary Noetherian Prime Rings and Idealizers
Description
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
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