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Some results on Cozero-divisor graph of a commutative ring

Akbari, S ; Sharif University of Technology

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  1. Type of Document: Article
  2. DOI: 10.1142/S0219498813501132
  3. Abstract:
  4. 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 non-zero and non-unit elements of R, and two distinct vertices a and b are adjacent if and only if a ∉ Rb and b ∉ Ra. In this paper, we show that if Γ′(R) is a forest, then Γ′(R) is a union of isolated vertices or a star. Also, we prove that if Γ′(R) is a forest with at least one edge, then R ≅ ℤ2 ⊕ F, where F is a field. Among other results, it is shown that for every commutative ring R, diam(Γ′(R[x])) = 2. We prove that if R is a field, then Γ′(R[[x]]) is totally disconnected. Also, we prove that if (R, m) is a commutative local ring and m ≠ 0, then diam(Γ′(R[[x]])) ≤ 3. Finally, it is proved that if R is a commutative non-local ring, then diam(Γ′(R[[x]])) ≤ 3
  5. Keywords:
  6. Cozero-divisor graph ; Diameter
  7. Source: Journal of Algebra and its Applications ; Vol. 13, issue. 3 , May , 2014 ; ISSN: 02194988
  8. URL: http://www.worldscientific.com/doi/abs/10.1142/S0219498813501132