On the planarity of the kk-zero-divisor hypergraphs

Let RR be a commutative ring with identity and let Z(R,k)Z(R,k) be the set of all kk-zero-divisors in RR and k>2k>2 an integer. The kk-zero-divisor hypergraph   of RR, denoted by Hk(R)Hk(R), is a hypergraph with vertex set Z(R,k)Z(R,k), and for distinct element x1,x2,…,xkx1,x2,…,xk in Z(R,k)Z(R,k), the set {x1,x2,…,xk}{x1,x2,…,xk} is an edge of Hk(R)Hk(R) if and only if x1x2⋯xk=0x1x2⋯xk=0 and the product of elements of no (k−1)(k−1)-subset of {x1,x2,…,xk}{x1,x2,…,xk} is zero. In this paper, we characterize all finite commutative non-local rings RR for which the kk-zero-divisor hypergraph is planar.