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

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.