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Hereditary noetherian prime rings and idealizers

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...

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Detalles Bibliográficos
Autores principales: Levy, Lawrence S, Robson, J Chris
Lenguaje:eng
Publicado: American Mathematical Society 2011
Materias:
Acceso en línea:http://cds.cern.ch/record/2623034
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author Levy, Lawrence S
Robson, J Chris
author_facet Levy, Lawrence S
Robson, J Chris
author_sort Levy, Lawrence S
collection CERN
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 describes some open problems. The history of the topics is surveyed at appropriate points.
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spelling cern-26230342021-04-21T18:47:53Zhttp://cds.cern.ch/record/2623034engLevy, Lawrence SRobson, J ChrisHereditary noetherian prime rings and idealizersMathematical Physics and MathematicsThe 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.American Mathematical Societyoai:cds.cern.ch:26230342011
spellingShingle Mathematical Physics and Mathematics
Levy, Lawrence S
Robson, J Chris
Hereditary noetherian prime rings and idealizers
title Hereditary noetherian prime rings and idealizers
title_full Hereditary noetherian prime rings and idealizers
title_fullStr Hereditary noetherian prime rings and idealizers
title_full_unstemmed Hereditary noetherian prime rings and idealizers
title_short Hereditary noetherian prime rings and idealizers
title_sort hereditary noetherian prime rings and idealizers
topic Mathematical Physics and Mathematics
url http://cds.cern.ch/record/2623034
work_keys_str_mv AT levylawrences hereditarynoetherianprimeringsandidealizers
AT robsonjchris hereditarynoetherianprimeringsandidealizers