A GENERALIZATION OF ZERO-DIVISOR GRAPH

Let R be a commutative ring with identity, Z(R) be the set of zero-divisors of R, and let Γ(R) be the zero-divisor graph of R. In this talk, we introduce and study a generalization of zero-divisor graphs. This new graph associated with R is defined as the graph Γ g (R) with the vertex set Z(R) ∗ = Z(R) n f 0 g , and two distinct vertices x and y are adjacent if and only if ann R (x) + ann R (y) is an essential ideal of R. It follows that Γ g (R) contains Γ(R) as a subgraph. The diameter, the girth and conditions under which Γ g (R) equals to Γ(R) are discussed.