A general theory of zero-divisor graphs over a commutative ring

Abstract

Let R be a commutative ring with 1 ≠ 0, I a proper ideal of R, and ~ a multiplicative congruence relation on R. Let R/~ = { [x]~ | x ∈ R} be the commutative monoid of ~-congruence classes under the induced multiplication [x]~[y]~ = [xy]~, and let Z(R/~) be the set of zero-divisors of R/~. The ~-zero-divisor graph of R is the (simple) graph Γ~(R) with vertices Z(R/~) \ {[0]~} and with distinct vertices [x]~ and [y]~ adjacent if and only if [x]~[y]~ = [0]~. Special cases include the usual zero-divisor graphs Γ(R) and Γ(R/I), the ideal-based zero-divisor graph ΓI (R), and the compressed zero-divisor graphs ΓE(R) and ΓE(R/I). In this paper, we investigate the structure and relationship between the various ~-zero-divisor graphs.