Lecture Notes on Algebraic Structure of Lattice-Ordered Rings.pdf
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.
Partially Ordered Sets; Lattices; Lattice-Ordered Groups; Vector Lattices; Lattice-Ordered Rings and Algebras; Lattice-Ordered Algebras with a d-Basis; Positive Derivations on Lattice-Ordered Rings; Recognition of Lattice-Ordered Matrix Rings with the Entrywise Order; Positive Cycles; Nonzero f-Elements in Lattice-Ordered Rings; Quotient Rings of Lattice-Ordered Ore Domains; Lattice-Ordered Matrix Algebras with Totally Ordered Integral Domains; d-Elements That Are Not Positive; Lattice-Ordered Triangular Matrix Rings; l-Ideals of Lattice-Ordered Rings with Positive Identity.