Zero Divisor Graph of a Commutative Ring

Abstract

The main aim is to relate the theoretic properties of a commutative ring with properties of graph. For a commutative ring, the set of zero-divisors of denoted by Z(R). A simple graph Г(R) is associated with the vertices which are non zero zero-divisors denoted by Z(R)*=Z(R)-{0}, where for distinct non zero zero-divisors of R x, y,  the vertices x and y are connected by an edge if x y=0.This study illustrates the structure of Г(R)  and the properties of Z(R). We study when Г(R) can be a complete graph and a star graph and examine the connectivity and diameter and grith of the graph Г(R). We also study Г(R) for non-isomorphic rings. The properties of Г(R), for a commutative ring R and If Z(R) is an annihilator ideal, and for a local ring R with maximal ideal M are given.