Non-Commutative Valuation Rings and Semi-Hereditary Orders by H. MarubayashiNon-Commutative Valuation Rings and Semi-Hereditary Orders by H. Marubayashi

Non-Commutative Valuation Rings and Semi-Hereditary Orders

byH. Marubayashi, Haruo Miyamoto, Akira Ueda

Paperback | December 7, 2010

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Much progress has been made during the last decade on the subjects of non­ commutative valuation rings, and of semi-hereditary and Priifer orders in a simple Artinian ring which are considered, in a sense, as global theories of non-commu­ tative valuation rings. So it is worth to present a survey of the subjects in a self-contained way, which is the purpose of this book. Historically non-commutative valuation rings of division rings were first treat­ ed systematically in Schilling's Book [Sc], which are nowadays called invariant valuation rings, though invariant valuation rings can be traced back to Hasse's work in [Has]. Since then, various attempts have been made to study the ideal theory of orders in finite dimensional algebras over fields and to describe the Brauer groups of fields by usage of "valuations", "places", "preplaces", "value functions" and "pseudoplaces". In 1984, N. 1. Dubrovin defined non-commutative valuation rings of simple Artinian rings with notion of places in the category of simple Artinian rings and obtained significant results on non-commutative valuation rings (named Dubrovin valuation rings after him) which signify that these rings may be the correct def­ inition of valuation rings of simple Artinian rings. Dubrovin valuation rings of central simple algebras over fields are, however, not necessarily to be integral over their centers.
Title:Non-Commutative Valuation Rings and Semi-Hereditary OrdersFormat:PaperbackDimensions:192 pages, 27.9 × 21 × 0.01 inPublished:December 7, 2010Publisher:Springer-Verlag/Sci-Tech/TradeLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:9048148537

ISBN - 13:9789048148530


Table of Contents

Preface. I. Semi-Hereditary and Prüfer Orders. II. Dubrovin Valuation Rings. III. Semi-Local Bezout Orders. IV. The Applications and Examples. Appendix: A1. Semi-Perfect Rings and Serial Rings. A2. Coherent Rings. A3. Azumaya Algebras. A4. The Lifting Idempotents. A5. Wedderburn's Theorem. References. Index of Notation. Index.