On rings whose prime ideal sum graphs are line graphs

TL;DR

This paper characterizes commutative Artinian rings based on whether their prime ideal sum graphs are line graphs or complements of line graphs, linking ring structure to graph-theoretic properties.

Contribution

It provides a complete characterization of Artinian rings with prime ideal sum graphs as line graphs or their complements, connecting algebraic and graph-theoretic concepts.

Findings

Characterization of rings with prime ideal sum graphs as line graphs.

Description of rings whose prime ideal sum graphs are complements of line graphs.

Insight into the structure of Artinian rings via graph properties.

Abstract

Let $R$ be a commutative ring with unity. The prime ideal sum graph of the ring $R$ is the simple undirected graph whose vertex set is the set of all nonzero proper ideals of $R$ and two distinct vertices $I$, $J$ are adjacent if and only if $I + J$ is a prime ideal of $R$. In this paper, we characterize all commutative Artinian rings whose prime ideal sum graphs are line graphs. Finally, we give a description of all commutative Artinian rings whose prime ideal sum graph is the complement of a line graph.