Near Rings, Fuzzy Ideals and Graph Theory by Bhavanari Satyanarayana and Kuncham Syam Prasad

Preface

Near rings, fuzzy ideals, and graph theory is a very fascinating course. Near ring theory has enormous applications in different subject areas such as digital computing, sequential mechanics, automata theory, graph theory, and combinatorics. The frst step toward near rings was axiomatic research done by Dickson in 1905. He exhibited that there do exist “felds with only one distributive law.” Near rings arise in a natural way. The set M(G) of all mappings of a group (G, ) into itself, with the usual addition and composition of mappings becomes a near ring. Another example is the set of all polynomials with addition and substitution.

This book provides the reader with a comprehensive idea about near ring theory with some links to fuzzy ideals and graph theory. It broadly covers three major topics: near rings, fuzzy ideals, and graph theory. Chapter 1 discusses all necessary fundamentals of algebraic systems. Chapters 2 and 3 cover the essentials of the fundamentals of near rings theory, appropriate examples, notations, and simple theorems. Chapter 4 covers the prime ideal concept in near rings. The rigorous approach of the dimension theory of N-groups, along with suitable illustrations, is presented in Chapter 5. Most of the chapters cover topics from the recent literature about certain notions and results such as prime ideals, essential ideals, uniform ideals, fnite dimension, and primary and tertiary decompositions along with several characterizations. In Chapter 6, we make a brief study of matrix near rings with detailed proofs presented wherever necessary. The concept of gamma near ring, a generalization of the concepts of both gamma and near rings, is presented in Chapter 7 with suitable results. In Chapter 8, an introduction to fuzzy algebraic systems is presented. Particularly, the fuzzy ideals of near rings and gamma near rings are studied extensively. In Chapter 9, an attempt is made to discuss the concept of graph theory. A good presentation on some important concepts such as directed hypercubes, dimension, prime graphs, and graphs with respect to ideals in near rings completes the book.

This textbook can be used by students as an introductory course for MSc (pure/applied mathematics) and as a prescribed book for MPhil and PhD course work. In addition, one can select suitable parts from this textbook to frame the syllabus of the relevant papers for different courses according to one’s needs