Classification of rings with toroidal co-annihilating-ideal graphs

Let R be a commutative ring with identity. The co-annihilating-ideal graph of R, denoted by A(R), is a graph whose vertex set is the set of all nonzero proper ideals of R and two distinct vertices I and J are adjacent whenever Ann(I) boolean AND Ann(J) = {0}. In this paper, we study the planarity and genus of A(R). In particular, we characterize all Artinian rings R for which the genus of A(R) is zero or one.