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

TL;DR

This paper investigates the properties of nil-graphs of ideals in commutative rings, focusing on conditions for completeness, bipartiteness, independence number, and genus classification of Artinian rings.

Contribution

It provides new characterizations of nil-graphs of ideals, including conditions for their completeness, bipartiteness, and genus, and determines the independence number for reduced rings.

Findings

Characterizes when nil-graphs are complete or bipartite.

Determines the independence number for nil-graphs of reduced rings.

Classifies Artinian rings with nil-graphs of genus at most one.

Abstract

Let $R$ be a commutative ring with identity and ${\rm Nil}(R)$ 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 $\{I:\ (0)\neq I\lhd R$ and there exists a non-trivial ideal $J$ such that $IJ\subseteq {\rm Nil}(R)\}$ and two distinct vertices $I$ and $J$ are adjacent if and only if $IJ\subseteq {\rm Nil}(R)$. 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.