Central Sets and Radii of the Zero-Divisor Graphs of Commutative Rings

Abstract

For a commutative ring R with identity, the zero-divisor graph, Γ(R), is the graph with vertices the nonzero zero-divisors of R and edges between distinct vertices x and y whenever xy = 0. This article gives a proof that the radius of Γ(R) is 0, 1, or 2 if R is Noetherian. The center union {0} is shown to be a union of annihilator ideals if R is Artinian. The diameter of Γ(R) can be determined once the center is identified. If R is finite, then the median is shown to be a subset of the center. A dominating set of Γ(R) is constructed using elements of the center when R is Artinian. It is shown that for a finite ring R ≇ ℤ₂ × F for some finite field F, the domination number of Γ(R) is equal to the number of distinct maximal ideals of R. Other results on the structure of Γ(R) are also presented.

Key Words:

• Central sets
• Commutative ring
• Zero-divisor
• Zero-divisor graph

Mathematics Subject Classification:

• Primary 13A99
• Secondary 05C99, 13M99

ACKNOWLEDGMENTS

The author wishes to thank everyone associated with the DIMACS Satellite Program hosted by the Illinois Institute of Technology in July 2003 for an introduction to the topic of centrality in a graph. Gratitude is also extended to the University Research Council of Eastern Kentucky University for providing funds that enabled the author to complete this article.

Notes

Communicated by J. Kuzmanovich.