Lattice-ordered Rings and Modules
This book provides an exposition of the algebraic aspects of the theory of lattice-ordered rings and lattice-ordered modules. All of the background material on rings, modules, and lattice-ordered groups necessary to make the work self-contained and&n
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Lattice-ordered Rings and Modules
Lattice-ordered Rings and Modules
Lattice-ordered Rings and Modules
Stuart A. Steinberg Toledo, OH, USA
Stuart A. Steinberg Department of Mathematics University of Toledo Toledo, OH 43601 USA [email protected]
ISBN 978-1-4419-1720-1 e-ISBN 978-1-4419-1721-8 DOI 10.1007/978-1-4419-1721-8 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2009940319 Mathematics Subject Classification (2010): 06F25, 13J25, 16W60, 16W80, 06F15, 12J15, 13J05, 13J30, 12D15 © Springer Science+Business Media, LLC 2010 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
To Diane Stephen, David, and Julia
Preface
A lattice-ordered ring is a ring that is also a lattice in which each additive translation is order preserving and the product of two positive elements is positive. Many ring constructions produce a ring that can be lattice-ordered in more than one way. This text is an account of the algebraic aspects of the theories of lattice-ordered rings and of those lattice-ordered modules which can be embedded in a product of totally ordered modules—the f -modules. It is written at a level which is suitable for a second-year graduate student in mathematics, and it can serve either as a text for a course in lattice-ordered rings or as a monograph for a researcher who wishes to learn about the subject; there are over 800 exercises of various degrees of difficulty which appear at the ends of the sections. Included in the text is all of the relevant background information that is needed in order to to make the theories that are developed and the results that are presented comprehensible to readers with various backgrounds. In order to make this book as self-contained as possible it was necessary to include a large amount of background material. Thus, in the first chapter we have constructed the Dedekind and MacNeille completions of a partially ordered set (poset) and developed enough of universal algebra so that we can present Birkhoff’s characterization of a variety and so that we can also verify the existence of free objects in a variety of algebras. Much of the material on lattice-ordered groups (`-groups) in the second chapter appears in those books devoted to the subject. What is new in this boo
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