On the annihilator-ideal graph of commutative rings

Let R be a commutative ring with nonzero identity. We denote by AG(R) the annihilator graph of R, whose vertex set consists of the set of nonzero zero-divisors of R, and two distint vertices x and y are adjacent if and only if ann(x) boolean OR ann(y) not equal ann(xy), where for t is an element of R, we set ann(t) := {r is an element of R vertical bar rt = 0}. In this paper, we define the annihiator-ideal graph of R, which is denoted by A(I)(R), as an undirected graph with vertex set A*(R), and two distinct vertices I and J are adjacent if and only if ann(I)boolean OR ann(J)not equal ann(IJ). We study some basic properties of A(I)(R) such as connectivity, diameter and girth. Also we investigate the situations under which the graphs AG(R) and A(I)(R) are coincide. Moreover, we examin the planarity of the graph A(I)(R).