Classifying annihilating-ideal graphs of commutative artinian rings

ABSTRACT

In this article we investigate the annihilating-ideal graph of a commutative ring, introduced by Behboodi and Rakeei in [10]. 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 six or fewer vertices which can be realized as the annihilating-ideal graph of a commutative ring.

KEYWORDS:

• Annihilating-ideal graph
• commutative artinian rings
• Quasi-Frobenius rings

MATHEMATICS SUBJECT CLASSIFICATION:

• 05C25
• 13A15

Acknowledgments

The authors are extremely grateful to the anonymous referee, whose careful reading and thoughtful comments led to marked improvements.