Characterization of rings with genus two prime ideal sum graphs

TL;DR

This paper characterizes finite non-local commutative rings whose prime ideal sum graphs have genus two, providing a classification based on the structure of the rings and their ideals.

Contribution

It offers a complete characterization of rings with prime ideal sum graphs of genus two, a novel classification in ring and graph theory.

Findings

Identifies all finite non-local rings with genus two prime ideal sum graphs

Provides structural conditions for rings based on their prime ideal sum graphs

Advances understanding of the relationship between ring structure and graph genus

Abstract

Let $R$ be a commutative ring with unity. The prime ideal sum graph of the ring $R$ is a simple undirected graph whose vertex set is the set of nonzero proper ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and only if $I + J$ is a prime ideal of $R$. In this paper, we characterize all the finite non-local commutative rings whose prime ideal sum graph is of genus $2$.