On the Genus of Strong Annihilating-ideal Graph of Commutative Rings

Let R be a commutative ring with unity and A(R) be the set of annihilating-ideals of R. The strong annihilating-ideal graph of R, denoted by SAG(R), is an undirected graph with vertex set A(R)* and two vertices I-1 and I-2 are adjacent if and only if I-1 boolean AND Ann(I-2) not equal (0) and I-2 boolean AND Ann(I-1) not equal (0). In this paper, first we characterize commutative Artinian rings whose strong annihilating-ideal graph is isomorphic to some well-known graphs and then we classify commutative Artinian rings whose strong annihilating-ideal graph is planar, toroidal or projective.