On the extended graph associated with the set of all non-zero annihilating ideals of a commutative ring

Abstract

Let Rbe a commutative ring with non-zero identity which is not an integral domain. An ideal I of a ring R is called an annihilating ideal if there exists r∈R\{0} such that Ir=(0). Let A(R) denote the set of all annihilating ideals of R and A (R)*=A(R)\{0}. In this article, we introduce a new graph associated with R denoted by H(R) whose vertex set is A(R)* and two distinct vertices I, J are adjacent in this graph if and only if IJ=(0) or I+J ∈ A(R). The aim of this article is to study the interplay between the ring-theoretic properties of a ring R and the graph-theoretic properties of H(R). For such a ring R, we prove that H(R) is connected and find its diameter. Moreover, we determine girth of H(R). Furthermore, we provide some sufficient conditions under which H(R) is a complete graph.