The weakly annihilating-ideal graph of a commutative ring

Abstract

Let R be 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 and investigate the weakly annihilating-ideal graph of R denoted by WAG(R). It is the undirected graph whose vertex set is A(R)^∗ and two distinct vertices I, J are adjacent in this graph if and only if there exist non-zero ideals A, B of R with A ⊆ ann(I) and B ⊆ ann(J) such that AB=(0). The aim of this article is to study the interplay between the ring-theoretic properties of R and the graph-theoretic properties of WAG(R). We discuss some results regarding the connectedness of WAG(R) and determine its diameter and girth. Moreover, we provide some conditions under which WAG(R) and AG(R) are identical.