Beschreibung:
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. 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.
Steinberg presents the material through 800+ extensive examples of varying levels of difficulty along with numerous exercises at the end of each section. 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.
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. 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. This text may be used as a textbook or for self-study by graduate students and researchers studying lattice-ordered rings and lattice-ordered modules.
Clearly written and self-contained
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.-
