THE ANNIHILATING-IDEAL GRAPH OF A COMMUTATIVE RING WITH RESPECT TO AN IDEAL

For a commutative ring R with identity, the annihilating-ideal graph of R, denoted ??(R), is the graph whose vertices are the nonzero annihilating ideals of R with two distinct vertices joined by an edge when the product of the vertices is the zero ideal. We will generalize this notion for an ideal I of R by replacing nonzero ideals whose product is zero with ideals that are not contained in I and their product lies in I and call it the annihilating-ideal graph of R with respect to I, denoted ??(I)(R). We discuss when ??(I)(R) is bipartite. We also give some results on the subgraphs and the parameters of ??(I)(R).