SOME CRITERIA FOR THE FINITENESS OF COZERO-DIVISOR GRAPHS

作者：Akbari S^*; Khojasteh S

来源：Journal of Algebra and Its Applications, 2013, 12(8): 1350056.

DOI：10.1142/S0219498813500564

• Cozero-divisor graph
• domination number
• independence number

Let R be a commutative ring with unity. The cozero-divisor graph of R denoted by Gamma%26apos; (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 is not an element of Rb and b is not an element of Ra, where Rc is the ideal generated by the element c in R. Let alpha(Gamma%26apos; (R)) and gamma(Gamma%26apos; (R)) denote the independence number and the domination number of Gamma%26apos; (R), respectively. In this paper, we prove that if alpha(Gamma%26apos; (R)) is finite, then R is Artinian if and only if R is Noetherian. Also, we prove that if alpha(Gamma%26apos; (R)) is finite, then R/P is finite, for every prime ideal P. Moreover, we prove that if R is a Noetherian ring, gamma(Gamma%26apos; (R)) is finite and Gamma%26apos; (R) has at least one isolated vertex, then J(R) = N(R). We show that if R is a commutative Noetherian local ring, gamma(Gamma%26apos; (R)) is finite and Gamma%26apos; (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 Gamma%26apos; (R) is finite and positive, then R is a finite ring.

• 出版日期2013-12