ON THE COZERO-DIVISOR GRAPHS AND COMAXIMAL GRAPHS OF COMMUTATIVE RINGS

Abstract

Let R be a commutative ring with non-zero identity. The cozero-divisor graph of R, denoted by Γ′(R), is a graph with vertices in W*(R), which is the set of all non-zero and non-unit elements of R, and two distinct vertices a and b in W*(R) are adjacent if and only if a ∉ bR and b ∉ aR. Also the comaximal graph is a graph with vertices all elements of R and two distinct vertices a and b are adjacent if and only if aR + bR = R. We denote the subgraph of the comaximal graph with vertex-set W*(R), by Γ2(R). In this note, we study the relations between two graphs Γ′(R) and Γ2(R). Also, we investigate the cozero-divisor graphs of idealizations of commutative rings.