Communications in Combinatorics and Optimization (Jun 2017)
Some results on the complement of a new graph associated to a commutative ring
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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