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Paper IPM / M / 7809  


Abstract:  
Let R be a commutative ring with nonzero identity and
let I be an ideal of R. The zerodivisor graph of R with
respect to I, denoted by Γ_{I}(R), is the graph whose
vertices are the set {x ∈ R\I xy ∈ I forsome y ∈ R\I} with distinct vertices x and y
adjacent if and only if xy ∈ I. In the case I=0,
Γ_{0}(R), denoted by Γ(R), is the zerodivisor graph
which has well known results in the literature. In this article we
explore the relationship between Γ_{I}(R) ≅ Γ_{J}(S) and Γ(R/I) ≅ Γ(S/J). We also discuss when Γ_{I}(R) is bipartite. Finally we give some
results on the subgraphs and the parameters of Γ_{I}(R).
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