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Let R be a commutative ring with identity and let Ω(R) * be the set of all nontrivial principal ideals of R. The reduced cozero-divisor graph Γ r (R) of R is an undirected simple graph with Ω(R) * as the vertex set and two distinct vertices (x) and (y) in Ω(R) * are adjacent if and only if (x) (y) and (y) (x). In this paper, we characterize all classes of commutative Artinian non-local rings for which the reduced cozero-divisor graph has genus at most one.

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