On some embedding of the strong annihilating-ideal graph of commutative rings

Let S be a commutative ring with unity (CRU) and W(S) be the set of annihilating-ideals of S. The strong annihilating-ideal graph of S, denoted by SAG(S), is an undirected graph with vertex set W(S)*. Two vertices m and n are adjacent if and only if m ? Ann(n) ? (0) and n ? Ann(m) ? (0). In this paper, we first characterize the Artinian commutative rings S for which SAG(S) has outerplanarity index 2. Then, we classify Artinian commutative rings S for which SAG(S) is double toroidal or Klein-bottle. Finally, we determine the book thickness of SAG(S) for genus at most one.