Near Rings, Fuzzy Ideals, and Graph Theory: 1st Edition (Hardback) book cover

Near Rings, Fuzzy Ideals, and Graph Theory

1st Edition

By Bhavanari Satyanarayana, Kuncham Syam Prasad

Chapman and Hall/CRC

480 pages | 111 B/W Illus.

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Hardback: 9781439873106
pub: 2013-05-21
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Description

Near Rings, Fuzzy Ideals, and Graph Theory explores the relationship between near rings and fuzzy sets and between near rings and graph theory. It covers topics from recent literature along with several characterizations.

After introducing all of the necessary fundamentals of algebraic systems, the book presents the essentials of near rings theory, relevant examples, notations, and simple theorems. It then describes the prime ideal concept in near rings, takes a rigorous approach to the dimension theory of N-groups, gives some detailed proofs of matrix near rings, and discusses the gamma near ring, which is a generalization of both gamma rings and near rings. The authors also provide an introduction to fuzzy algebraic systems, particularly the fuzzy ideals of near rings and gamma near rings. The final chapter explains important concepts in graph theory, including directed hypercubes, dimension, prime graphs, and graphs with respect to ideals in near rings.

Near ring theory has many applications in areas as diverse as digital computing, sequential mechanics, automata theory, graph theory, and combinatorics. Suitable for researchers and graduate students, this book provides readers with an understanding of near ring theory and its connection to fuzzy ideals and graph theory.

Reviews

"The book is quite self-contained in that the introduction describes the basic concepts covered in the rest of the book, including some notions on fuzzy sets and fuzzy algebraic groups. … The vast majority of the results, even preliminary ones, are accompanied by proofs, making the book quite interesting for readers who want to get acquainted with the theory of near rings."

Computing Reviews, October 2013

"The authors present the theory of near rings, and conclude with two chapters on the application to fuzzy ideals and graph theoretical investigations in near rings. … The book is quite self-contained, in that the introduction describes the basic concepts covered in the rest of the book, including some notions on fuzzy sets and fuzzy algebraic groups. The content is organized in a standard way, with numbering of definitions, lemmas, examples, and notes. Most of the definitions and results are accompanied by references to the literature, mainly to previous results produced by the authors or to other books written by some of the authors. The vast majority of the results, even preliminary ones, are accompanied by proofs, making the book quite interesting for readers who want to get acquainted with the theory of near rings. Since most of the book is really about near rings, the target audience includes readers that are interested in near rings and in ring properties that transfer to near rings and their connection to geometry."

—Marcelo Finger, Computing Reviews, October 2013

Table of Contents

Preliminaries

Set Theory

Group Theory

Ring Theory

Vector Spaces

Module Theory

Graphs

Introduction to Fuzzy Sets

Some Fuzzy Algebraic Systems

Fundamentals of Near Rings

Definitions and Examples

Substructures of Near Rings and Quotient Near Rings

Homomorphism and Isomorphism

Introduction to Matrix Near Rings

More Concepts on Near Rings

N-Groups

Homomorphisms in N-Groups

Chain Conditions on Ideals

Direct Systems in N-Groups

Some Results on Mn(N)-Group Nn

Prime and Semiprime Ideals

Fundamental Definitions and Results

Prime Radicals in Near Rings

Insertion of Factors Property Ideals in Near Rings and N-Groups

Finite Spanning Dimension in N-Groups

Dimension and Decomposition Theory

Finite Goldie Dimension

Linearly Independent Elements in N-Groups

Primary Decomposition in Near Rings

Tertiary Decomposition in N-Groups

Matrix Near Rings

Preliminary Results

Prime Ideals in Matrix Near Rings

Finite Goldie Dimension in Mn(N)–Group Nn

Gamma Near Rings

Definitions and Examples

Ideals of Gamma Near Rings

Prime Ideals and Nilpotent Ideals in Gamma Near Rings

Modules over Gamma Near Rings

Fuzzy Aspects in Near Rings and Gamma Near Rings

Fuzzy Ideals and Prime Ideals in Gamma Near Rings

Fuzzy Cosets of Gamma Near Rings

Fuzzy Ideals of MΓ-Groups

Homomorphisms and Fuzzy Cosets of MΓ-Groups

Fuzzy Dimension in N-Groups

Fundamental Concepts of Graph Theory

Introduction

Directed Hypercubes

Vector Spaces of a Graph

Prime Graph of a Ring

Homomorphism of Graphs

Graph of a Near Ring with Respect to an Ideal

References

Index

About the Originator

Subject Categories

BISAC Subject Codes/Headings:
COM046000
COMPUTERS / Operating Systems / General
MAT002000
MATHEMATICS / Algebra / General
MAT036000
MATHEMATICS / Combinatorics