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

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