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On the ideal-based zero-divisor graph of commutative rings

K. Selvakumar

Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli, Tamil Nadu 627012, India

E-mail Address: selva_158@yahoo.co.in

Corresponding author.

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and

N. Anusha

Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli, Tamil Nadu 627012, India

E-mail Address: anusha9320n@gmail.com

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https://doi.org/10.1142/S1793830922501907Cited by:0 (Source: Crossref)

Abstract

Let $R$ $R$ be a finite commutative ring with identity, $I$ $I$ be an ideal of $R$ $R$ and $J (R)$ $J (R)$ denotes the Jacobson radical of $R$ $R$ . The ideal-based zero-divisor graph $Γ_{I} (R)$ $Γ_{I} (R)$ of $R$ $R$ is a graph with vertex set $V (Γ_{I} (R)) = {x \in R - I : x y \in I,$ $V (Γ_{I} (R)) = {x \in R - I : x y \in I,$ for some $y \in R - I}$ $y \in R - I}$ in which distinct vertices $x$ $x$ and $y$ $y$ are adjacent if and only if $x y \in I$ $x y \in I$ . In this paper, we determine the diameter, girth of $Γ_{J (R)} (R)$ $Γ_{J (R)} (R)$ . Specifically, we classify all finite commutative nonlocal rings for which $Γ_{J (R)} (R)$ $Γ_{J (R)} (R)$ is perfect. Furthermore, we discuss about the planarity, outerplanarity, genus and crosscap of $Γ_{J (R)} (R)$ $Γ_{J (R)} (R)$ and characterize all of them.

Communicated by Xiao-Dong Zhang

Keywords:

AMSC: 05C50, 05C12, 15A18

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