Some Properties of the Nil-Graphs of Ideals of Commutative Rings

Abstract

Let $R$ be a commutative ring with identity and $\mathrm{Nil}(R)$ be the set of nilpotent elements of $R$. The nil-graph of ideals of $R$ is defined as the graph ${\mathbb{AG}}_N(R)$ whose vertex set is $\{I:(0)\ne I⊲R$ and there exists a non-trivial ideal $J$ such that $IJ\subseteq \mathrm{Nil}(R)\}$ and two distinct vertices $I$ and $J$ are adjacent if and only if $IJ\subseteq \mathrm{Nil}(R)$. Here, we study conditions under which ${\mathbb{AG}}_N(R)$ is complete or bipartite. Also, the independence number of ${\mathbb{AG}}_N(R)$ is determined, where $R$ is a reduced ring. Finally, we classify Artinian rings whose nil-graphs of ideals have genus at most one.

13 pages