On the extended cozero-divisor graph

Let R be a commutative ring with 1≠0 and W∗(R) its set of nonzero nonunits. In this paper, we introduce and study a new graph that is closely related to the cozero-divisor graph Γ′(R). We define the extended cozero-divisor graph of R to be the (simple) graph Γ′¯(R) with vertices W∗(R), and distinct vertices x and y are adjacent if and only if there are positive integers m and n such that xm∉ynR and yn∉xmR. We determine when Γ′(R)=Γ′¯(R) and show that Γ′¯(R) is a complete graph if and only if aR={0,a} for every a∈W∗(R). We also study the diameter and girth of Γ′¯(R) and show that gr (Γ′¯(R1×R2))=3 if and only if either R1 is not a valuation ring, R2 is not a valuation ring, or R1 and R2 are not fields, and that gr (Γ′(R))= gr (Γ′¯(R))∈{3,4,∞} when R is a commutative Artinian ring. Many examples are given to illustrate the theory along with some new results for Γ′(R).