Translator Disclaimer
October 2017 The extended zero-divisor graph of a commutative ring II
M. BAKHTYIARI, M. J. NIKMEHR, R. NIKANDISH
Hokkaido Math. J. 46(3): 395-406 (October 2017). DOI: 10.14492/hokmj/1510045304

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.

Citation

Download Citation

M. BAKHTYIARI. M. J. NIKMEHR. R. NIKANDISH. "The extended zero-divisor graph of a commutative ring II." Hokkaido Math. J. 46 (3) 395 - 406, October 2017. https://doi.org/10.14492/hokmj/1510045304

Information

Published: October 2017
First available in Project Euclid: 7 November 2017

zbMATH: 06814869
MathSciNet: MR3720335
Digital Object Identifier: 10.14492/hokmj/1510045304

Subjects:
Primary: 05C10, 05C69, 13B99

Rights: Copyright © 2017 Hokkaido University, Department of Mathematics

JOURNAL ARTICLE
12 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.46 • No. 3 • October 2017
Back to Top