some remarks on the annihilating-ideal graph of commutative ring with respect to an ideal

The graph ag(r) of a commutative ring r with identity has an edge link- ing two unique vertices when the product of the vertices equals the zero ideal and its vertices are the nonzero annihilating ideals of r. the annihilating- ideal graph with respect to an ideal (i), which is denoted by agi (r), has distinct vertices k and j that are adjacent if and only if kj ⊆ i. its vertices are k kj i for some ideal j and k, j