Commutative Rings Whose Zero-Divisor Graphs Have Positive Genus

Abstract

It was shown by C. Wickham [12] that “for a fixed positive integer g, there are finitely many isomorphism classes of finite commutative rings whose zero-divisor graph has genus g.” In this note, we give a short direct proof for this result. Moreover, we show that, if the zero-divisor graph of a commutative ring R has finite genus g, then either g = 0 or R is a finite ring. This immediately generalizes Wickham's theorem to arbitrary (not necessary finite) commutative rings.

Key Words:

• Commutative ring
• Genus of a graph
• Zero-divisor graph

2010 Mathematics Subject Classification:

• 05C25
• 13H99
• 13M99

ACKNOWLEDGMENTS

The authors wish to express their deep appreciation to the referee who carefully read an earlier version of this paper and made significant suggestions for improvement. The research of the second author was in part supported by a grant from IPM (No. 90160034).

Notes

Communicated by T. Albu.