On the genus of graphs from commutative rings

Let R be a commutative ring with non-zero identity. The cozero-divisor graph of R, denoted by Γ′(R), is a graph with vertex-set W∗(R), which is the set of all non-zero non-unit elements of R, and two distinct vertices x and y in W∗(R) are adjacent if and only if x∉Ry and y∉Rx, where for z∈R, Rz is the ideal generated by z. In this paper, we determine all isomorphism classes of finite commutative rings R with identity whose Γ′(R) has genus one. Also we characterize all non-local rings for which the reduced cozero-divisor graph Γr(R) is planar