Classification of nonlocal rings with genus one 3-zero-divisor hypergraphs. Issue 1 (2nd January 2017) - Selvakumar, K.; Ramanathan, V.
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 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 ∏ i = 1 k 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.