WHEN IDEAL-BASED ZERO-DIVISOR GRAPHS ARE COMPLEMENTED OR UNIQUELY COMPLEMENTED

Abstract

Let R be a commutative ring with nonzero identity and I a proper ideal of R. The ideal-based zero-divisor graph of R with respect to the ideal I, denoted by ΓI (R), is the graph on vertices {x ∈ R \ I | xy ∈ I for some y ∈ R\I}, where distinct vertices x and y are adjacent if and only if xy ∈ I. In this paper, we give a complete classification of when an ideal-based zero-divisor graph of a commutative ring is complemented or uniquely complemented based on the total quotient ring of R/I.