ON GENERALIZED ZERO-DIVISOR GRAPH ASSOCIATED WITH A COMMUTATIVE RING

Abstract

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 Γ_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 Γ(R) is an edge (path) of Γ_g(R).  It is proved that Γ_g(R) is connected with diameter at most three and with girth at most four, if Γ_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.