[内容简介]

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.

[目录]

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 M_n(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 M_n(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