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 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. 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.' Front Matter....Pages 1-15 Partially Ordered Sets and Lattices....Pages 1-31 Lattice-ordered Groups....Pages 33-123 Lattice-ordered Rings....Pages 125-279 The Category of f -Modules....Pages 281-417 Lattice-ordered Fields....Pages 419-509 Additional Topics....Pages 511-607 Back Matter....Pages 1-21