## Hokkaido Mathematical Journal

### The extended zero-divisor graph of a commutative ring II

#### Abstract

Let $R$ be a commutative ring with identity, and let $Z(R)$ be the set of zero-divisors of $R$. The extended zero-divisor graph of $R$ is the undirected (simple) graph $\Gamma'(R)$ with the vertex set $Z(R)^*=Z(R)\setminus\{0\}$, and two distinct vertices $x$ and $y$ are adjacent if and only if either $Rx\cap \mathrm{Ann}(y)\neq (0)$ or $Ry\cap \mathrm{Ann}(x)\neq (0)$. In this paper, we continue our study of the extended zero-divisor graph of a commutative ring that was introduced in [4]. We show that the extended zero-divisor graph associated with an Artinian ring is weakly perfect, i.e., its vertex chromatic number equals its clique number. Furthermore, we classify all rings whose extended zero-divisor graphs are planar.

#### Article information

Source
Hokkaido Math. J., Volume 46, Number 3 (2017), 395-406.

Dates
First available in Project Euclid: 7 November 2017

https://projecteuclid.org/euclid.hokmj/1510045304

Digital Object Identifier
doi:10.14492/hokmj/1510045304

Mathematical Reviews number (MathSciNet)
MR3720335

Zentralblatt MATH identifier
06814869

#### Citation

BAKHTYIARI, M.; NIKMEHR, M. J.; NIKANDISH, R. The extended zero-divisor graph of a commutative ring II. Hokkaido Math. J. 46 (2017), no. 3, 395--406. doi:10.14492/hokmj/1510045304. https://projecteuclid.org/euclid.hokmj/1510045304