Lecture Notes On Algebraic Structure Of Lattice-ordered Rings

Algebraic Structure of Lattice-Ordered Rings presents an introduction to the theory of lattice-ordered rings and some new developments in this area in the last 10-15 years. It aims to provide the reader with a good foundation in the subject, as well as some new research ideas and topic in the field.

This book may be used as a textbook for graduate and advanced undergraduate students who have completed an abstract algebra course including general topics on group, ring, module, and field. It is also suitable for readers with some background in abstract algebra and are interested in lattice-ordered rings to use as a self-study book.

The book is largely self-contained, except in a few places, and contains about 200 exercises to assist the reader to better understand the text and practice some ideas.
Contents:Introduction to Ordered Algebraic Systems:LatticesLattice-Ordered Groups and Vector LatticesLattice-Ordered Rings and AlgebrasLattice-Ordered Algebras with a d-Basis:Examples and Basic PropertiesStructure TheoremsPositive Derivations on ℓ-Rings:Examples and Basic Propertiesƒ-Ring and Its GeneralizationsMatrix ℓ-RingsKernel of a Positive DerivationSome Topics on Lattice-Ordered Rings:Recognition of Matrix ℓ-Rings with the Entrywise OrderPositive CyclesNonzero ƒ-Eelements in ℓ-RingsQuotient Rings of Lattice-Ordered Ore DomainsMatrix ℓ-Algebras Over Totally Ordered Integral Domainsd-Elements That are Not PositiveLattice-Ordered Triangular Matrix Algebrasℓ-Ideals of ℓ-Unital Lattice-Ordered Rings:Maximal ℓ-Idealsℓ-Ideals in commutative ℓ-Unital ℓ-Rings
Readership: Graduate students in algebra and number theory.
Key Features:The book includes new material such as positive derivations on lattice-ordered rings, lattice-ordered triangular matrix algebrasMore details are provided in proofs of the results in the book for beginners to understand the textThe book presents new research ideas, methods and topics suitable to advanced undergraduate students and master students