ON GENERALIZED ZERO-DIVISOR GRAPH ASSOCIATED WITH A COMMUTATIVE RING

Let R be a commutative ring with identity, and let Z(R) be the set of zero-divisors of R. The generalized zero-divisor graph of R is defined as the graph Gamma(g)(R) with the vertex set Z(R)* = Z(R) \ {0}, 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 each edge (path) of the zero-divisor graph G(R) is an edge (path) of Gamma(g)(R). It is proved that Gamma(g)(R) is connected with diameter at most three and with girth at most four, if Gamma(g)(R) contains a cycle. Furthermore, all rings with the same generalized zero-divisor and zero-divisor graphs are characterized. Among other results, we show that the generalized zero-divisor graph associated with an Artinian ring is weakly perfect, i.e., its vertex chromatic number equals its clique number.