Lattice-Ordered Rings and Modules

ISBN-13: 9781441917201 / Angielski / Twarda / 2009 / 630 str.

Stuart A. Steinberg

Opis

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 accessible to a variety of readers is included. Steinberg includes in his presentation of the material 800+ extensive exercises of varying levels of difficulty at the end of each of the sections. The first two chapters of the book provide a thorough introduction to the material, while the following four chapters delve into more specific topics. Key topics include: *lattice-ordered groups, rings, and fields; *archimedean $l$-groups; *f-rings and larger varieties of $l$-rings; *the category of f-modules; *various commutativity results. Filling a gap in the literature, Lattice-Ordered Rings and Modules may be used as a textbook or for self-study by graduate students and researchers studying lattice-ordered rings and lattice-ordered modules.

Recenzje

From the reviews:

"Lattice-ordered rings and modules are of interest both to algebraists and to functional analysts. Previous authors have presented only those results that are needed for the applications that they have in mind. Having a book like this one would be a great boon. Both the algebraists and the analysts would benefit from having the fundamental results all gathered together in one book. This book would fill in the gap admirably.

Any algebraist or functional analyst whose work involves ordered structures would be interested in this book. That includes mathematicians who study: lattices, ordered semigroups, ordered groups, ordered rings, ordered fields, Riesz spaces, vector lattices, or Banach lattices. It would also be useful for mathematicians who are interested in general ring theory, general group theory, matrix algebras, or function algebras (especially Banach algebras). In short, there is a wide audience who would find the tools that this book provides very useful."

"Lattice-ordered rings and modules is the latest in a line of books aimed at simultaneously detailing the history while up-dating the canon of the theory of lattice-ordered groups and rings. ... The author has been able to create a book that is both accessible to graduate students and useful to researchers. ... the material is presented in a clear and well-written manner ... every researcher in the areas of lattice-ordered groups and rings will come to possess a copy of Lattice-ordered rings and modules." (Warren Wm. McGovern, Mathematical Reviews, Issue 2011 b)

"This book gives a comprehensive account of the algebraic aspects of the theories of lattice-ordered rings and those lattice-ordered modules which can be embedded in a product of totally ordered modules, the f-modules. It is written at a level suitable for gifted second-year graduate students in mathematics and ... serve as a monograph for researchers in these fields. ... A list of 27 open problems concludes the book, which will be the text of reference to lattice-ordered rings and modules for many years to come." (H. Mitsch, Monatshefte für Mathematik, Vol. 162 (3), March, 2011)

"This is an excellent book on lattice-ordered groups, rings and, especially, modules. ... there are more than 800 extensive exercises in the book. Another advantage of the book is that at the end of every chapter there is a useful Note which provides the references used in the chapter. Finally, the book ends with 27 interesting open problems." (Yichuan Yang, Zentralblatt MATH, Vol. 1205, 2011)

Spis treści

Preface.- List of Symbols.- 1 Partially Ordered Sets and Lattices.- 1.1 Partially Ordered Sets.- 1.2 Lattices.- 1.3 Completion.- 1.4 Universal Algebra.- 2 Lattice-ordered Groups.- 2.1 Basic Identities and Examples.- 2.2 Subobjects and Homomorphisms.- 2.3 Archimedean `-groups.- 2.4 Prime Subgroups, Representability, and Operator Sets.- 2.5 Values.- 2.6 Hahn Products and the Embedding Theorem.- 3 Lattice-ordered Rings.- 3.1 Basics, Examples, and Nonexamples.- 3.2 Radical Theory.- 3.3 f -Rings.- 3.4 Embedding in a Unital f -Algebra.- 3.5 Generalized Power Series Rings.- 3.6 Archimedean f -Rings.- 3.7 Squares Positive.- 3.8 Polynomial Constraints.- 4 The Category of f -Modules.- 4.1 Rings of Quotients and Essential Extensions.- 4.2 Torsion Theories and Rings of Quotients.- 4.3 Lattice-ordered Rings and Modules of Quotients.- 4.4 Injective f -Modules.- 4.5 Free f -Modules.- 5 Lattice-ordered Fields.- 5.1 Totally Ordered Extensions of Ordered Fields.- 5.2 Valuations and the Hahn Embedding Theorem.- 5.3 Lattice-ordered Fields.- 6 Additional Topics.- 6.1 Lattice-ordered Semigroup Rings.- 6.2 Algebraic f -Elements Are Central.- 6.3 More Polynomial Constraints on Totally Ordered Domains.- 6.4 Lattice-ordered Matrix Algebras.- Open Problems.- References.- Index.-