A GENERAL THEORY OF ZERO-DIVISOR GRAPHS OVER A COMMUTATIVE RING

Year 2016, , 111 - 135, 01.12.2016

David F. Anderson Elizabeth F. Lewis

https://doi.org/10.24330/ieja.266187

Cited By: 6

Abstract

Let R be a commutative ring with 1 6= 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.

Keywords

Zero-divisor, zero-divisor graph, ideal-based zero-divisor graph, compressed zero-divisor graph, congruence-based zero-divisor graph