Perfectness of a graph associated with annihilating ideals of a ring

Let R be a commutative ring with identity which is not an integral domain. An ideal I of a ring R is called an annihilating ideal if there exists r is an element of R\{0} such that Ir = (0). Let Omega(R) be a simple undirect graph associated with R whose vertex set is the set of all nonzero annihilating ideals of R and two distinct vertices I, J are joined if and only if I+J is also an annihilating ideal of R. A perfect graph is a graph in which the chromatic number of every induced subgraph equals the size of the largest clique of that subgraph. In this paper, we characterize all rings whose Omega(R) is perfect.