Finite commutative rings whose line graphs of comaximal graphs have genus at most two

Let R be a ring with identity. The comaximal graph of R, denoted by Γ(R), is a simple graph with vertex set R and two different vertices a and b are adjacent if and only if $aR+bR=R$. Let Γ₂(R) be a subgraph of Γ(R) induced by R(R)∪ J(R)\. In this paper, we investigate the genus of the line graph L(Γ(R)) of Γ(R) and the line graph L(Γ₂(R)) of Γ₂(R). All finite commutative rings whose genus of L(Γ(R)) and L(Γ₂(R)) are 0, 1, 2 are completely characterized, respectively.