Embedding of prime ideal sum graph of a commutative ring on surfaces

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 classify non-local commutative rings $R$ such that $\text{PIS}(R)$ is of crosscap at most two. We prove that there does not exist a finite non-local commutative ring whose prime ideal sum graph is projective planar. Further, we classify non-local commutative rings of genus one prime ideal sum graphs. Moreover, we classify finite non-local commutative rings for which the prime ideal sum graph is split graph, threshold graph, cograph, cactus graph and unicyclic, respectively