ON THE ANNIHILATOR INTERSECTION GRAPH OF A COMMUTATIVE RING

Abstract

Let R be a commutative ring with identity and A(R) be the set of ideals with non-zero annihilator.  The annihilator intersection graph of R is defined as the graph \(\mathbb{AIG}\)(R) with the vertex set A(R)^∗ = A(R) \ {0} and two distinct vertices I and J are adjacent if and only if Ann(IJ) ≠ Ann(I) ∩ Ann(J).  It follows that the annihilating-ideal graph \(\mathbb{AG}\)(R) (a well-known graph with the same vertices and two distinct vertices I, J are adjacent if and only if IJ = 0) is a subgraph of \(\mathbb{AIG}\)(R). It is proved that \(\mathbb{AIG}\)(R) is connected with diameter at most two and with girth at most four, if \(\mathbb{AIG}\)(R) contains a cycle.  Moreover, we characterize all rings whose annihilator intersection graphs are complete or star.  Furthermore, we study the affinity between annihilator intersection graph and annihilating-ideal graph associated with a ring.