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.

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 Γr(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.Mathematics Subject Classification (2000) 05C10 · 05C25 · 05C75