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WINTER 2013 When is the complement of the zero-divisor graph of a commutative ring planar?
S. Visweswaran
J. Commut. Algebra 5(4): 567-601 (WINTER 2013). DOI: 10.1216/JCA-2013-5-4-567

Abstract

Let $R$ be a commutative ring with identity admitting at least two distinct zero-divisors $a,b$ with $ab\neq 0$. In this article, necessary and sufficient conditions are determined in order that $(\Gamma(R))^{c}$ (that is, the complement of the zero-divisor graph of $R$) is planar. It is noted that, if $(\Gamma(R))^{c}$ is planar, then the number of maximal $N$-primes of $(0)$ in $R$ is at most three. Firstly, we consider rings $R$ admitting exactly three maximal $N$-primes of $(0)$ and present a characterization of such rings in order that the complement of their zero-divisor graphs be planar. Secondly, we consider rings $R$ admitting exactly two maximal $N$-primes of $(0)$ and investigate the problem of when the complement of their zero-divisor graphs is planar. Thirdly, we consider rings $R$ admitting only one maximal $N$-prime of $(0)$ and determine necessary and sufficient conditions in order that the complement of their zero-divisor graphs be planar.

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S. Visweswaran. "When is the complement of the zero-divisor graph of a commutative ring planar?." J. Commut. Algebra 5 (4) 567 - 601, WINTER 2013. https://doi.org/10.1216/JCA-2013-5-4-567

Information

Published: WINTER 2013
First available in Project Euclid: 31 January 2014

zbMATH: 1299.13006
MathSciNet: MR3161747
Digital Object Identifier: 10.1216/JCA-2013-5-4-567

Subjects:
Primary: 13A15

Keywords: maximal $N$-primes of (0) , The complement of the zero-divisor graph

Rights: Copyright © 2013 Rocky Mountain Mathematics Consortium

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Vol.5 • No. 4 • WINTER 2013
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