The k -Zero-Divisor Hypergraph of a Commutative Ring

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 .