On line upper ideal relation graphs of rings

TL;DR

This paper characterizes Artinian rings based on the structure of their upper ideal relation graphs, specifically identifying when these graphs are line graphs or their complements.

Contribution

It provides a complete characterization of Artinian rings whose upper ideal relation graphs are line graphs or their complements.

Findings

Identifies all Artinian rings with upper ideal relation graphs as line graphs.

Describes all Artinian rings where these graphs are complements of line graphs.

Abstract

The upper ideal relation graph $\Gamma_{U}(R)$ of a commutative ring $R$ with unity is a simple undirected graph with the set of all non-unit elements of $R$ as a vertex set and two vertices $x$, $y$ are adjacent if and only if the principal ideals $(x)$ and $(y)$ are contained in the principal ideal $(z)$ for some non-unit element $z\in R$. This manuscript characterizes all the Artinian rings $R$ such that the graph $\Gamma_{U}(R)$ is a line graph. Moreover, all the Artinian rings $R$ for which $\Gamma_{U}(R)$ is the complement of a line graph have been described.