Ch. Eslahchi¹ and A. M. Rahimi²

Copyright © 2007 Ch. Eslahchi and A. M. Rahimi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.