Classification of nonlocal rings with genus one 3-zero-divisor hypergraphs

Abstract

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 elements x1,x2,…,xk in Z(R,k), the set {x1,x2,…,xk} is an edge of ℋk(R) if and only if ∏i=1kxi=0 and the product of any (k−1) elements of {x1,x2,…,xk} is nonzero. In this paper, we characterize all finite commutative nonlocal rings R with identity whose ℋ3(R) has genus one.