Some Remarks on the Graph Γ I (R)

Abstract

Let R be a commutative ring with nonzero identity and Z(R) its set of zero-divisors. The zero-divisor graph of R is Γ(R), with vertices Z(R)∖{0} and distinct vertices x and y are adjacent if and only if xy = 0. For a proper ideal I of R, the ideal-based zero-divisor graph of R is Γ I (R), with vertices {x ∈ R∖I | xy ∈ I for some y ∈ R∖I} and distinct vertices x and y are adjacent if and only if xy ∈ I. In this article, we study the relationship between the two graphs Γ(R) and Γ I (R). We also determine when Γ I (R) is either a complete graph or a complete bipartite graph and investigate when Γ I (R) ≅ Γ(S) for some commutative ring S.