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Paper   IPM / M / 7809
School of Mathematics
  Title:   Zero-divisor graph with respect to an ideal
1.  H. R. Maimani
2.  M. R. Pournaki
3.  S. Yassemi
  Status:   Published
  Journal: Comm. Algebra
  Vol.:  34
  Year:  2006
  Pages:   923-929
  Supported by:  IPM
Let R be a commutative ring with nonzero identity and let I be an ideal of R. The zero-divisor graph of R with respect to I, denoted by ΓI(R), is the graph whose vertices are the set {xR\IxyI forsome yR\I} with distinct vertices x and y adjacent if and only if xyI. In the case I=0, Γ0(R), denoted by Γ(R), is the zero-divisor graph which has well known results in the literature. In this article we explore the relationship between ΓI(R) ≅ ΓJ(S) and Γ(R/I) ≅ Γ(S/J). We also discuss when ΓI(R) is bipartite. Finally we give some results on the subgraphs and the parameters of ΓI(R).

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