Annihilator Ideal-Based Zero-Divisor Graphs Over Multiplication Modules

Abstract

For a commutative ring R with identity, an ideal-based zero-divisor graph, denoted by Γ _I (R), is the graph whose vertices are {x ∈ R∖I | xy ∈ I for some y ∈ R∖I}, and two distinct vertices x and y are adjacent if and only if xy ∈ I. In this article, we investigate an annihilator ideal-based zero-divisor graph by replacing the ideal I with the annihilator ideal Ann(M) for a multiplication R-module M. Based on the above-mentioned definition, we examine some properties of an R-module over a von Neumann regular ring, and the cardinality of an R-module associated with Γ _Ann(M)(R).

Key Words:

• Annihilator ideal-based zero-divisor graphs
• Multiplication modules
• Von Neumann regular rings

2010 Mathematics Subject Classification:

• 13A99
• 05C99
• 13C99

ACKNOWLEDGMENTS

The authors wish to acknowledge Professor David F. Anderson for his helpful comments.

Notes

Communicated by I. Swanson.