On the planarity of the k-zero-divisor hypergraphs

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