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

Authors : David F Anderson, Elizabeth F Lewis

Pages : 111-135

Doi:10.24330/ieja.266187

View : 11 | Download : 12

Publication Date : 2016-12-01

Article Type : Research Paper

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 Zinsert ignore into journalissuearticles values(R/∼); be the set of zero-divisors of R/∼. The ∼-zero-divisor graph of R is the insert ignore into journalissuearticles values(simple); graph Γ∼insert ignore into journalissuearticles values(R); with vertices Zinsert ignore into journalissuearticles values(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 Γinsert ignore into journalissuearticles values(R); and Γinsert ignore into journalissuearticles values(R/I);, the ideal-based zero-divisor graph ΓI insert ignore into journalissuearticles values(R);, and the compressed zero-divisor graphs ΓEinsert ignore into journalissuearticles values(R); and ΓEinsert ignore into journalissuearticles values(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