The k -Zero-Divisor Hypergraph of a Commutative Ring

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 H k ( 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 H k ( R ) . It is shown that if R has two prime ideals P 1 and P 2 with zero their only common point, then H k ( R ) is a bipartite ( 2 -colorable) hypergraph with partition sets P 1 − Z ′ and P 2 − 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 H 3 ( R ) is found. Also, we have shown that H 3 ( R ) is connected with diameter at most 4 whenever x 2 ≠0 for all 3 -zero-divisors x of R . Finally, it is shown that for any finite nonlocal ring R , the hypergraph H 3 ( R ) is complete if and only if R is isomorphic to Z 2 × Z 2 × Z 2 .