Generalizations of the Zero-Divisor Graph of a Ring

Let R be a commutative ring with 1, and let Z(R) denote the set of zero­divisors of R. We define an undirected graph Γ(R) with vertices Z(R)* = Z(R) - {0}, where distinct vertices x and y of R are connected if and only if xy = 0. This graph is called the zero-divisor graph of R. We extend the definition of the zero-divisor graph to a noncommutative ring in several ways. Next, given a commutative ring R and ideal Ι of R, we introduce the notion of an ideal-based graph. This is an undirected graph with vertex set {x ∈ R – Ι| xy ∈ Ι for some y ∈ R - Ι}, where distinct vertices x and y are adjacent if and only if xy ∈ Ι. The properties of such a graph are investigated. We give several results concerning the zero-divisor graph of a commutative ring. Finally, the appendix gives examples illustrating an equivalence relation on the vertices of Γ(R) that can be used to produce a related graph for rings R of specific types.