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May 30 , 2008

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.

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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.

  • Introduction to Ordered Algebraic Systems:
    • Lattices
    • Lattice-Ordered Groups and Vector Lattices
    • Lattice-Ordered Rings and Algebras
  • Lattice-Ordered Algebras with a d-Basis:
    • Examples and Basic Properties
    • Structure Theorems
  • Positive Derivations on ℓ-Rings:
    • Examples and Basic Properties
    • ƒ-Ring and Its Generalizations
    • Matrix ℓ-Rings
    • Kernel of a Positive Derivation
  • Some Topics on Lattice-Ordered Rings:
    • Recognition of Matrix ℓ-Rings with the Entrywise Order
    • Positive Cycles
    • Nonzero ƒ-Eelements in ℓ-Rings
    • Quotient Rings of Lattice-Ordered Ore Domains
    • Matrix ℓ-Algebras Over Totally Ordered Integral Domains
    • d-Elements That are Not Positive
    • Lattice-Ordered Triangular Matrix Algebras
  • ℓ-Ideals of ℓ-Unital Lattice-Ordered Rings:
    • Maximal ℓ-Ideals
    • ℓ-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 algebras
  • More details are provided in proofs of the results in the book for beginners to understand the text
  • The book presents new research ideas, methods and topics suitable to advanced undergraduate students and master students
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