2School of Mathematics, University of Tehran, Tehran-IRAN
Abstract
The purpose of this paper is the study of zero-divisor graphs of a commutative multiplicative hyperrings, as a generalization of commutative rings. In this regards we consider a commutative multiplicative hyperring (π ,+,π), where (π ,+) is an abelian group, (π ,+) is a semihypergroup and for all π,π,π β π , :π π (π + π) β π π π + π π π and (π + π) π π β π π π + π π π. For π β π a nonzero element π β π is said to be a zero-divisor of π, if 0 β π π π and the set of zero-divisors of π is denoted by π(π ). We associative to π a zero-divisor graph π€(π ), whose vertices of π€(π ) are the elements of π(π )β(= Z(R)\{0}) and two distinct vertices of π€(π ) are adjacent if they were in π(π ). Finally, we obtain some properties of π€(π ) and compare some of its properties to the zero-divisor graph of a classical commutative ring and show that almost all properties of zero-divisor graphs of a commutative ring can be extend to π€(π ) while π is a strongly distributive multiplicative hyperring.