some results on domination in annihilating-ideal graphs of commutative rings

Let r be a commutative ring with identity and a(r) be the set of all ideals of r with non-zero annihilators. the annihilating-ideal graph of r is defined as the graph ag(r) with the vertex set a∗(r) = a(r){(0)} and two distinct vertices i and j are adjacent if and only if ij = (0). let g = (v; e) be a graph. a domination set for g is a subset s of v such that every vertex not in s is joined to at least one member of s by some edge. the domination number γ(g) is the minimum  cardinality among the dominating sets of g. in this paper, we study and characterize the dominating sets and domination numbers of the annihilating-ideal graph ag(r) for a commutative ring r.