Characterization of rings with planar, toroidal or projective planar prime ideal sum graphs

TL;DR

This paper explores the relationship between algebraic properties of rings and the topological features of their prime ideal sum graphs, classifying rings based on planarity, toroidality, and genus.

Contribution

It classifies non-local commutative Artinian rings with prime ideal sum graphs of specific topological types, including crosscap and genus one classifications.

Findings

No non-local Artinian ring has a projective planar prime ideal sum graph.

Classified rings with prime ideal sum graphs of crosscap at most two.

Identified conditions for rings with genus one prime ideal sum graphs.

Abstract

Let $R$ be a commutative ring with unity. The prime ideal sum graph $\text{PIS}(R)$ 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 $I$ and $J$ are adjacent if and only if $I + J$ is a prime ideal of $R$. In this paper, we study some interplay between algebraic properties of rings and graph-theoretic properties of their prime ideal sum graphs. In this connection, we classify non-local commutative Artinian rings $R$ such that $\text{PIS}(R)$ is of crosscap at most two. We prove that there does not exist a non-local commutative Artinian ring whose prime ideal sum graph is projective planar. Further, we classify non-local commutative Artinian rings of genus one prime ideal sum graphs.