Loading...

Commutative rings whose cozero-divisor graphs are unicyclic or of bounded degree

Akbari, S ; Sharif University of Technology

240 Viewed
  1. Type of Document: Article
  2. DOI: 10.1080/00927872.2012.745867
  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 vertex set W*(R), where W*(R) is the set of all nonzero and nonunit 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. Recently, it has been proved that for every nonlocal finite ring R, Γ′(R) is a unicyclic graph if and only if R ≅ ℤ2 × ℤ4, ℤ3 × ℤ3, ℤ2 × ℤ2[x]/(x 2). We generalize the aforementioned result by showing that for every commutative ring R, Γ′(R) is a unicyclic graph if and only if R ≅ ℤ2 × ℤ4, ℤ3 × ℤ3, ℤ2 × ℤ2[x]/(x 2), ℤ2[x, y]/(x, y)2, ℤ4[x]/(2x, x 2). We prove that for every positive integer Δ, the set of all commutative nonlocal rings with maximum degree at most Δ is finite. Also, we classify all rings whose cozero-divisor graph has maximum degree 3. Among other results, it is shown that for every commutative ring R, gr(Γ′(R)) ∈ {3, 4, ∞}
  5. Keywords:
  6. Cozero-divisor graph ; Girth ; Unicyclic graph
  7. Source: Communications in Algebra ; Vol. 42, Issue. 4 , 2014 , pp. 1594-1605 ; ISSN: 0092-7872
  8. URL: http://www-tandfonline-com.access.ezproxy.ir/doi/abs/10.1080/00927872.2012.745867#.Vb8U7C6Hi-E