Irreducible Divisor Graphs in Commutative Rings with Zero Divisors

Abstract

In their article “Irreducible Divisor Graphs”, Coykendall and Maney (2007) introduced the idea of irreducible divisor graphs of elements of a domain. We generalize this concept to commutative rings with zero divisors. In particular, the interplay of unique factoring and connected/complete graphs is explored. The diameter and girth of such graphs are also briefly discussed.

Key Words:

• Commutative ring
• Irreducible graph

2000 Mathematics Subject Classification:

• Primary 13A99
• Secondary 13A05

ACKNOWLEDGMENTS

The authors would like to thank the referee for many helpful comments that significantly improved this article.

Notes

Communicated by I. Swanson.