Compressed intersection annihilator graph

Abstract

Let R be a commutative ring with a non-zero identity. In this paper, we define a new graph, the compressed intersection annihilator graph, denoted by IA(R), and investigate some of its properties and its relation with the structure of the ring. It is a generalization of the torsion graph ΓR(R) and has some advantages over the torsion graph and some other graphs. We study classes of rings for which the equivalence between the set of zero-divisors of R, Z(R), being an ideal and the completeness of IA(R) holds. We also study the relation between ΓR(R) and IA(R). Besides, we show that if the compressed intersection annihilator graph of a ring R is finite, then there exists a subring S of R such that IA(S) ∼= IA(R). Also, we show that the compressed intersection annihilator graph will never be a complete bipartite graph. Besides, we show that the graph IA(R) with at least three vertices is connected and its diameter is less than or equal to three. Finally, we determine the properties of the graph in the cases when R is the ring of integers modulo n, the direct product of integral domains, the direct product of Artinian local rings and the direct product of two rings such that one of them is not an integral domain.