Cayley Graphs of Ideals in a Commutative Ring

M. Afkhami ; M. R. Ahmadi ; R. Jahani-Nezhad ; K. Khashyarmanesh

Bulletin of the Malaysian Mathematical Society, Tome 37 (2014) no. 3

Résumé

Let $R$ be a commutative ring. We associate a digraph to the ideals of $R$ whose vertex set is the set of all nontrivial ideals of $R$ and, for every twodistinct vertices $I$ and $J$, there is an arc from $I$ to $J$, denoted by $I\rightarrow J$, whenever there exists a nontrivial ideal $L$ such that $J=IL$. We call this graph the ideal digraph of $R$ and denote it by $\overrightarrow{I\Gamma}(R)$. Also, for a semigroup $H$ and a subset $S$ of $H$, the Cayley graph $T{Cay}(H,S)$ of $H$ relative to $S$ is defined as the digraph with vertex set $H$ and edge set $E(H,S)$ consisting of those ordered pairs $(x,y)$ such that $y=sx$ for some $s\in S$. In fact the ideal digraph $\overrightarrow{I\Gamma}(R)$ is isomorphic to the Cayley graph $T{Cay}(\mathfrak{I}^*,\mathfrak{I}^*)$, where $\mathfrak{I}$ is the set of all ideals of $R$ and $\mathfrak{I}^*$ consists of nontrivial ideals. The undirected ideal (simple) graph of $R$, denoted by $I\Gamma(R)$, has an edge joining $I$ and $J$ whenever either $J=IL$ or $I=JL$, for some nontrivial ideal $L$ of $R$. In this paper, we study some basic properties of graphs $\overrightarrow{I\Gamma}(R)$ and $I\Gamma(R)$ such as connectivity, diameter, graph height, Wiener index and clique number. Moreover, we study the Hasse ideal digraph $\overrightarrow{H\Gamma}(R)$, which is a spanning subgraph of $\overrightarrow{I\Gamma}(R)$ such that for each two distinct vertices $I$ and $J$, there is an arc from $I$ to $J$ in $\overrightarrow{H\Gamma}(R)$ whenever $I\rightarrow J$ in $\overrightarrow{I\Gamma}(R)$, and there is no vertex $L$ such that $I\rightarrow L$ and $L\rightarrow J$ in $\overrightarrow{I\Gamma}(R)$.

Détail

Classification : 05C20, 05C69, 13A15