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Lattice-Ordered Rings and Modules 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 in- clude a large amount of background material. Thus, in the first chapter we have con- structed the Dedekind and MacNeille completions of a partially ordered set (poset) and developed enough of universal algebra so that we can present Birkhoff’s char- acterization 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 book is the emphasis on `-groups with operators. This allows for the common development of basic results about `-groups, f -rings, and f -modules. The Amitsur- Kurosh theory of radicals is developed for the class of `-rings in the second section of Chapter 2 . Still more background material is given in the first two sections of Chapter 4 where the injective hull of a module, the Utumi maximal right quotient ring, and the ring of quotients and the module of quotients with respect to a hered- itary torsion theory are constructed and studied for a ring which is not necessarily unital. Also, the Artin–Schrier theory of totally ordered fields is given in the first section of Chapter 5, and enough of the theory of valuations on a field is presented in the second section so that a complete proof of the Hahn embedding theorem for a well-conditioned commutative lattice-ordered domain can be given. vii viii Preface Chapters 3, 4, 5, and 6 constitute the heart of the book. While not every known result on the topics included is presented, enough is presented so as to make the text, by which I mean the exercises also, reasonably complete. The first section of Chap- ter 3 develops the basic theory of `-rings including the fact that canonically ordered matrix rings have no unital f -modules. Section 4 shows that the fundamental process of embedding an f -algebra in a unital f -algebra is more complicated than the anal- ogous embedding for algebras and cannot always be carried out. The fifth section shows how to construct power series type examples of `-rings and `-modules using a poset which is a partial semigroup and which is rooted in the sense that the set of upper bounds of each element is a chain. The basic structure of f -rings is given in the third section of Chapter 3 and some of the richer structure of archimedean f -rings is given in the sixth section. In the last two sections the structure of `-rings in other varieties is examined. The seventh section studies those `-rings that have squares positive and gives conditions on a partially ordered generalized semigroup for the lexicographically ordered semigroup ring to have this property. The last sec- tion considers those `-rings which satisfy polynomial constraints more general than that of squares being positive. One effect of these constraints is to coalesce the set of nilpotent elements into a subring or an ideal and to force an `-semiprime ring to lack nilpotent elements. Also included in this section is a proof of the commutativity of an archimedean almost f -ring. Chapter 4 concentrates on the category of f -modules. The most conclusive re- sults occur for a semiprime f -ring whose maximal right quotient ring is an f -ring extension and whose Boolean algebra of annihilators is atomic. In the third section necessary and sufficient conditions for the module (or ring) of quotients to be an f -module (or an f -ring) extension are given, and the structure of right self-injective f -rings is given. The unique totally ordered right self-injective ring that does not have an identity element is exhibited. The module and order theoretic properties that determine when a nonsingular f -module is relatively injective are given in the fourth section—there are no injectives in this category of f -modules. A useful rep- resentation of the free nonsingular f -module is given in the last section and the size of a disjoint set in a free f -module is determined. In a totally ordered field the set of values—those convex subgroups which are maximal with respect to not containing a given nonzero element—becomes a to- tally ordered group under the operation induced on it by multiplication in the field. A proof of the Hahn Embedding Theorem for totally ordered fields is given in the second section of Chapter 5; namely, a totally ordered field is embedded in a power series field where the exponents belong to this value group of the field and the co- efficients are real numbers. Also, a totally ordered division ring is embedded in a totally ordered division algebra over the reals. In the third section of Chapter 5 the Hahn Embedding Theorem is given for a lattice-ordered commutative domain which satisfies a finiteness condition, and another embedding theorem for a suitably con- ditioned `-field into a formal power series crossed product `-ring is given. Also, the theory of archimedean `-fields is presented and lattice orders other than the usual total order are constructed for the field of real numbers. Preface ix Chapter 6 begins with a characterization of the canonically ordered real semi- group `-algebra over a locally finite left cancellative semigroup, and semigroup `- rings in which squares are positive are studied in more detail. In the second section it is shown that in an `-algebra in which the nonzero f -elements are not zero-divisors each algebraic f -element is central. A complete description is also given of those rings which have the property that each partial order is contained in a total order. In the third section more commutativity theorems are presented. It is shown that a totally ordered domain which is co-`-simple and which has a positive semidefinite form with a nontrivial solution must be commutative. A similar conclusion giving the centrality of f -elements appears for an `-ring in which the commutators are suit- ably bounded. A proof of Artin’s solution to Hilbert’s 17th problem which is mainly dependent on the variety of f -rings generated by the real numbers is also included in this section. Lattice orders on the n£n matrix algebra over a totally ordered field are considered in the last section. When the field is archimedean or when n = 2 all of the lattice orders are described, and it is shown that in these cases the usual order is the only lattice order in which the identity element is positive. A few words about the method of referencing are in order. An exercise is ref- erenced by its number alone when the exercise occurs in the section in which it is referenced, and, otherwise, it is referenced by its number preceded by the chapter and section numbers in which it occurs. A reference to a theorem or a numbered line uses all three of its numbers. Paul Taylor’s package was used in preparing the diagrams in the text. I wish to thank Joanne Guttman and Shirley Michel for their splendid job of typing and prepa- ration of the manuscript.
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