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International Journal of Mathematics and Mathematical Sciences
Volume 2007, Article ID 50875, 15 pages
Research Article

The k-Zero-Divisor Hypergraph of a Commutative Ring

1Department of Mathematical Sciences, Shahid Beheshti University, P.O. Box 19834, Tehran, Iran
2School 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

Academic Editor: Dalibor Froncek

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.


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 P1Z and P2Z, 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 x20 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.