Thek-Zero-Divisor Hypergraph of a Commutative Ring

Ch Eslahchi,A M Rahimi

doi:10.1155/2007/50875

Abstract

The concept of the zero-divisor graph of a commutative ring has been studied by many authors, and thek-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. LetRbe a commutative ring andkan integer strictly larger than2. Ak-uniform hypergraphHk(R)with the vertex setZ(R,k), the set of allk-zero-divisors inR, is associated toR, where eachk-subset ofZ(R,k)that satisfies thek-zero-divisor condition is an edge inHk(R). It is shown that ifRhas two prime idealsP1andP2with zero their only common point, thenHk(R)is a bipartite (2-colorable) hypergraph with partition setsP1−Z′andP2−Z′, whereZ′is the set of all zero divisors ofRwhich are notk-zero-divisors inR. IfRhas a nonzero nilpotent element, then a lower bound for the clique number ofH3(R)is found. Also, we have shown thatH3(R)is connected with diameter at most 4 wheneverx2≠0for all3-zero-divisorsxofR. Finally, it is shown that for any finite nonlocal ringR, the hypergraphH3(R)is complete if and only ifRis isomorphic toZ2×Z2×Z2.

Highlights

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
It is shown that if R has two prime ideals P1 and P2 with zero their only common point, 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
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

Summary

Introduction

The notion of a zero-divisor graph Γ(R) of a commutative ring R was first introduced by Beck in [1] and was further investigated in [2], where the authors were interested in colorings of Γ(R), though their vertex set included the zero element. We assume distinctness of the elements in Definition 1.1 for k-zero-divisors in order to have a k-uniform hypergraph, for any fixed integer k ≥ 3. A k-uniform hypergraph H is called complete if every k-subset of the vertices is an edge of H.

Results

Conclusion