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Non-Commutative Valuation Rings and Semi-Hereditary Orders

  • Hidetoshi Marubayashi
  • Haruo Miyamoto
  • Akira Ueda

Part of the K-Monographs in Mathematics book series (KMON, volume 3)

Table of contents

  1. Front Matter
    Pages i-viii
  2. Hidetoshi Marubayashi, Haruo Miyamoto, Akira Ueda
    Pages 1-20
  3. Hidetoshi Marubayashi, Haruo Miyamoto, Akira Ueda
    Pages 21-78
  4. Hidetoshi Marubayashi, Haruo Miyamoto, Akira Ueda
    Pages 79-108
  5. Hidetoshi Marubayashi, Haruo Miyamoto, Akira Ueda
    Pages 109-173
  6. Back Matter
    Pages 174-192

About this book

Introduction

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.

Keywords

Matrix algebra commutative ring ring theory

Authors and affiliations

  • Hidetoshi Marubayashi
    • 1
  • Haruo Miyamoto
    • 2
  • Akira Ueda
    • 3
  1. 1.Department of MathematicsNaruto University of EducationNarutoJapan
  2. 2.Department of MathematicsAnan College of TechnologyAnanJapan
  3. 3.Department of MathematicsShimane UniversityMatsueJapan

Bibliographic information

  • DOI https://doi.org/10.1007/978-94-017-2436-4
  • Copyright Information Springer Science+Business Media B.V. 1997
  • Publisher Name Springer, Dordrecht
  • eBook Packages Springer Book Archive
  • Print ISBN 978-90-481-4853-0
  • Online ISBN 978-94-017-2436-4
  • Series Print ISSN 1386-2804
  • Buy this book on publisher's site