Prime Divisors and Noncommutative Valuation Theory
Classical valuation theory has applications in number theory and class field theory as well as in algebraic geometry, e.g. in a divisor theory for curves. But the noncommutative equivalent is mainly applied to finite dimensional skewfields. Re
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2059
Hidetoshi Marubayashi
Fred Van Oystaeyen
Prime Divisors and Noncommutative Valuation Theory
123
Hidetoshi Marubayashi Tokushima Bunri University Faculty of Science and Engineering Shido, Sanuki City, Kagawa, Japan
Fred Van Oystaeyen University of Antwerp Mathematics and Computer Science Antwerp, Belgium
ISBN 978-3-642-31151-2 ISBN 978-3-642-31152-9 (eBook) DOI 10.1007/978-3-642-31152-9 Springer Heidelberg New York Dordrecht London Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2012945522 Mathematics Subject Classification (2010): 16W40, 16W70, 16S38, 16H10, 13J20, 16T05 c Springer-Verlag Berlin Heidelberg 2012 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Introduction
The classical theory of valuation rings in fields has natural applications in algebraic geometry as well as in number theory, e.g. as local rings of nonsingular points of curves or, respectively, as prime localizations of rings of integers in number fields. In these applications the rings are Noetherian and the valuations are discrete, that is the value group is the additive group of the integers, Z. The fact that non-discrete valuation ring
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