## Journal of Commutative Algebra

### When is the complement of the zero-divisor graph of a commutative ring planar?

S. Visweswaran

#### 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.

#### Article information

Source
J. Commut. Algebra, Volume 5, Number 4 (2013), 567-601.

Dates
First available in Project Euclid: 31 January 2014

https://projecteuclid.org/euclid.jca/1391192657

Digital Object Identifier
doi:10.1216/JCA-2013-5-4-567

Mathematical Reviews number (MathSciNet)
MR3161747

Zentralblatt MATH identifier
1299.13006

Subjects
Primary: 13A15: Ideals; multiplicative ideal theory

#### Citation

Visweswaran, S. When is the complement of the zero-divisor graph of a commutative ring planar?. J. Commut. Algebra 5 (2013), no. 4, 567--601. doi:10.1216/JCA-2013-5-4-567. https://projecteuclid.org/euclid.jca/1391192657

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