A GENERAL THEORY OF ZERO-DIVISOR GRAPHS OVER A COMMUTATIVE RING
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,
- ISSN: 1306-6048
- Yayın Aralığı: Yılda 2 Sayı
- Başlangıç: 2007
- Yayıncı: Abdullah HARMANCI