An Ideal-Based Zero-Divisor Graph of a Commutative Ring

Abstract

For a commutative ring R with identity, the zero-divisor graph of R, denoted Γ(R), is the graph whose vertices are the non-zero zero-divisors of R with two distinct vertices joined by an edge when the product of the vertices is zero. We will generalize this notion by replacing elements whose product is zero with elements whose product lies in some ideal I of R. Also, we determine (up to isomorphism) all rings R such that Γ(R) is the graph on five vertices.

Key Words:

• Zero-divisor
• Graph
• Ideal-based

Acknowledgments