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 ℋ_k(R), is a hypergraph with vertex set Z(R,k), and for distinct elements $x_1,x_2,\dots ,x_k$ in Z(R,k), the set $\{x_1,x_2,\dots ,x_k\}$ is an edge of ℋ_k(R) if and only if ${\prod }_{i=1}^k{x}_i=0$ and the product of any (k−1) elements of $\{x_1,x_2,\dots ,x_k\}$ is nonzero. In this paper, we characterize all finite commutative nonlocal rings R with identity whose ℋ₃(R) has genus one.

KEYWORDS:

• Hypergraph
• incidence graph
• planar hypergraph
• toroidal graph
• zero-divisor graph

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 05C12
• 05C25
• 05C69
• 13A99

Acknowledgments

The authors are deeply grateful to the referee for careful reading of the manuscript and helpful suggestions. The work reported here is supported by the UGC Major Research Project (F. No. 42-8/2013(SR)) awarded to K. Selvakumar by the University Grants Commission, Government of India.