# Some results on the complement of a new graph associated to a commutative ring

Document Type : Original paper

Authors

Saurashtra University

Abstract

The rings considered in this article are commutative with identity which admit at least one nonzero proper ideal.  Let $R$ be a ring. We denote the collection of all ideals of $R$  by $\mathbb{I}(R)$  and $\mathbb{I}(R)\backslash \{(0)\}$ by $\mathbb{I}(R)^{*}$. Alilou et al. [A. Alilou, J. Amjadi and S.M. Sheikholeslami,  {\em A new graph associated to a commutative ring}, Discrete Math. Algorithm. Appl.  {\bf 8} (2016) Article ID: 1650029 (13 pages)] introduced and investigated a new graph associated  to $R$, denoted by   $\Omega_{R}^{*}$ which  is an undirected  graph whose vertex set is $\mathbb{I}(R)^{*}\backslash \{R\}$ and distinct vertices $I, J$ are joined by an edge  in this graph if and only if either $(Ann_{R}I)J = (0)$ or $(Ann_{R}J)I = (0)$. Several interesting theorems were proved on $\Omega_{R}^{*}$ in the aforementioned paper and they  illustrate the  interplay between the graph-theoretic properties of $\Omega_{R}^{*}$ and the ring-theoretic properties of $R$. The aim of this article is to investigate some properties of $(\Omega_{R}^{*})^{c}$, the complement of the new graph $\Omega_{R}^{*}$  associated  to $R$.

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