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

## Main Article Content

## 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 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.

## Article Details

*Gulf Journal of Mathematics*,

*13*(1), 15-24. Retrieved from https://gjom.org/index.php/gjom/article/view/924