Classifying Annihilating-Ideal Graphs of Commutative Artinian Rings

In this article we investigate the annihilating-ideal graph of a commutative ring, introduced by Behboodi and Rakeei in [BR11a]. Our main goal is to determine which algebraic properties of a ring are reflected in its annihilating-ideal graph. We prove that, for artinian rings, the annihilating-ideal graph can be used to determine whether the ring in question is a PIR or, more generally, if it is a dual ring. Moreover, with one trivial exception, the annihilating-ideal graph can distinguish between PIRs with different ideal lattices. In addition, we explore new techniques for classifying small annihilating-ideal graphs. Consequently, we completely determine the graphs with 6 or fewer vertices which can be realized as the annihilating-ideal graph of a commutative ring