On the genus of strong annihilating-ideal graph of commutative rings

Let [Formula: see text] be a commutative ring with unity and [Formula: see text] be the set of annihilating-ideals of [Formula: see text]. The strong annihilating-ideal graph of [Formula: see text], denoted by [Formula: see text], is an undirected graph with vertex set [Formula: see text] and two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] and [Formula: see text]. 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.