Zero-Divisor Graphs with respect to Ideals in Noncommutative Rings

Abstract

Let R be a commutative ring and I an ideal of R . The zero-divisor graph of R with respect to I , denoted Γ I ( R ), is the undirected graph whose vertex set is { x ∈ R ∖ I | x y ∈ I for some y ∈ R ∖ I } with two distinct vertices x and y joined by an edge when x y ∈ I . In this paper, we extend the definition of the ideal-based zero-divisor graph to noncommutative rings.