SOME CRITERIA FOR THE FINITENESS OF COZERO-DIVISOR GRAPHS

Abstract

Let R be a commutative ring with unity. The cozero-divisor graph of R denoted by Γ'(R) is a graph with the vertex set W*(R), where W*(R) is the set of all nonzero and non-unit elements of R, and two distinct vertices a and b are adjacent if and only if a ∉ Rb and b ∉ Ra, where Rc is the ideal generated by the element c in R. Let α(Γ'(R)) and γ(Γ'(R)) denote the independence number and the domination number of Γ'(R), respectively. In this paper, we prove that if α(Γ'(R)) is finite, then R is Artinian if and only if R is Noetherian. Also, we prove that if α(Γ'(R)) is finite, then R/P is finite, for every prime ideal P. Moreover, we prove that if R is a Noetherian ring, γ(Γ'(R)) is finite and Γ'(R) has at least one isolated vertex, then J(R) = N(R). We show that if R is a commutative Noetherian local ring, γ(Γ'(R)) is finite and Γ'(R) has at least one isolated vertex, then R is a finite ring. Among other results, we prove that if R is a commutative ring and the maximum degree of Γ'(R) is finite and positive, then R is a finite ring.