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Some criteria for the finiteness of cozero-divisor graphs

Akbari, S ; Sharif University of Technology | 2013

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  1. Type of Document: Article
  2. DOI: 10.1142/S0219498813500564
  3. Publisher: 2013
  4. Abstract:
  5. 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
  6. Keywords:
  7. Cozero-divisor graph ; Domination number ; Independence number
  8. Source: Journal of Algebra and its Applications ; Volume 12, Issue 8 , 2013 ; 02194988 (ISSN)
  9. URL: http://www.worldscientific.com/doi/abs/10.1142/S0219498813500564