^{1}Department of Mathematical Sciences, Shahid Beheshti University, P.O. Box 19834, Tehran, Iran ^{2}School of Mathematics, Institute for Studies in Theoretical Physics and Mathematics, P.O. Box 19395-5746, Tehran, Iran

Received 28 June 2006; Revised 15 October 2006; Accepted 27 February 2007

The concept of the zero-divisor graph of a commutative ring has been studied by many authors, and the k-zero-divisor hypergraph of a commutative ring is a nice abstraction of this concept. Though some of the proofs in this paper are long and detailed, any reader familiar with zero-divisors will be able to read through the
exposition and find many of the results quite interesting. Let R be a commutative ring and k an integer strictly larger than 2. A k-uniform hypergraph Hk(R) with the vertex set Z(R,k), the set of all k-zero-divisors in R, is associated to R, where each k-subset of Z(R,k) that satisfies the
k-zero-divisor condition is an edge in Hk(R). It is shown
that if R has two prime ideals P1 and P2 with zero their
only common point, then Hk(R) is a bipartite (2-colorable) hypergraph with partition sets P1−Z′ and P2−Z′, where Z′ is the set of all zero divisors of R which are not
k-zero-divisors in R . If R has a nonzero nilpotent
element, then a lower bound for the clique number of H3(R) is
found. Also, we have shown that H3(R) is connected with diameter at most 4 whenever x2≠0 for all 3-zero-divisors x of R. Finally, it is shown that for any finite nonlocal
ring R, the hypergraph H3(R) is complete if and only if R is isomorphic to Z2×Z2×Z2.