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-zerodivisor hypergraph of R, denoted by H-k ( R), is a hypergraph with vertex set Z( R, k), and for distinct element x(1), x(2),..., x(k) in Z( R, k), the set {x1, x2,..., xk} is an edge of H-k ( R) if and only if x(1)x(2) ... x(k) = 0 and the product of elements of no ( k-1)-subset of {x(1), x(2),..., x(k)} is zero. In this paper, we characterize all finite commutative non-local rings R for which the k-zero-divisor hypergraph is planar. (C) 2015 Kalasalingam University. Production and Hosting by Elsevier B.V.