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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 study some interplay between algebraic properties of rings andgraph-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-localcommutative Artinian ring whose prime ideal sum graph is projective planar.Further, we classify non-local commutative Artinian rings of genus one primeideal sum graphs.

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