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

摘要

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 H-k(R), is a hypergraph with vertex set Z(R,k), and for distinct elements x(1), x(2), ... , x(k) in Z(R,k), the set {x(1), x(2), ... , x(k)} is an edge of H-k(R) if and only if Pi(k)(i=1) x(i) = 0 and the product of any (k - 1) elements of {x(1), x(2), ... , x(k)} is nonzero. In this paper, we characterize all finite commutative nonlocal rings R with identity whose H-3(R) has genus one.