the annihilator-inclusion ideal graph of a commutative ring

let r be a commutative ring with non-zero identity. the annihilator-inclusion ideal graph of r, denoted by ξr, is a graph whose vertex set is the of all non-zero proper ideals of r and two distinct vertices i and j are adjacent if and only if either ann(i)⊆j or ann(j)⊆i. the purpose of this paper is to provide some basic properties of the graph ξr. in particular, shows that ξr is a connected graph with diameter at most three, and has girth 3 or ∞. furthermore, is determined all isomorphic classes of non-local artinian rings whose annihilator-inclusion ideal graphs have genus zero or one.