The Congruence-Based Zero-Divisor Graph

Let R be a commutative ring with nonzero identity and ~ a multiplicative congruence relation on R. Then, R/~ is a semigroup with multiplication [x][y] = [xy], where [x] is the congruence class of an element x of R. We define the congruence-based zero-divisor graph of R ito be the simple graph with vertices the nonzero zero-divisors of R/~ and with an edge between distinct vertices [x] and [y] if and only if [x][y] = [0]. Examples include the usual zero-divisor graph of R, compressed zero-divisor graph of R, and ideal-based zero-divisor graph of R. We study relationships among congruence-based zero-divisor graphs for various congruence relations on R. In particular, we study connections between ring-theoretic properties of R and graph-theoretic properties of congruence-based zero-divisor graphs for various congruence relations on R.