Some results on the complement of the annihilating ideal graph of a commutative ring

Rings considered in this article are commutative with identity which admit at least one nonzero annihilating ideal. For such a ring R, we determine necessary and sufficient conditions in order that the complement of its annihilating ideal graph is connected and also find its diameter when it is connected. We discuss the girth of the complement of the annihilating ideal graph of R and prove that it is either equal to 3 or infinity. We also present a necessary and sufficient condition for the complement of the annihilating ideal graph to be complemented.