Rings whose zero-divisor graphs have positive genus
type: Item
instance of: scholarly article
title: Rings whose zero-divisor graphs have positive genus
zbMATH Open document ID: 1163.13003
author: Cameron Wickham
published in: Journal of Algebra
publication date: 2009-04-21T00:00:00Z
review text: Let \(R\) be a commutative ring with non-zero unity. The concept of the graph of the zero divisors of \(R\) was first introduced by \textit{I. Beck} [see J. Algebra 116, No. 1, 208--226 (1988; Zbl 0654.13001)], where he was mainly interested in colorings. In his work all elements of the ring were vertices of the graph. \textit{D.~F.~Anderson} and \textit{P. S. Livingston} [J. Algebra 217, No. 2, 434--447 (1999; Zbl 0941.05062)] associated a graph \(\Gamma(R)\) to \(R\), with vertices \(\mathbb Z(R)\setminus\{0\}\) the set of non-zero zero-divisors of \(R\), and for distinct \(x,y\in\mathbb Z(R)\setminus\{0\}\), the vertices \(x\) and \(y\) are adjacent if and only if \(xy=0\). In this paper it is shown that for any positive integer \(g\), there are finitely many isomorphism classes of rings whose zero-divisor graph has genus \(g\).
reviewed by: Siamak Yassemi
Mathematics Subject Classification ID: 13A99
Mathematics Subject Classification ID: 05C75
zbMATH DE Number: 5545838
zbMATH Keywords: zero divisor graph
zbMATH Keywords: finite ring
zbMATH Keywords: genus
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2008.09.040
OpenAlex ID: W2018392735
cites work: When a zero-divisor graph is planar or a complete \(r\)-partite graph
cites work: On the zero-divisor graph of a commutative ring
cites work: The zero-divisor graph of a commutative ring. II.
cites work: The zero-divisor graph of a commutative ring
cites work: Planar zero-divisor graphs
cites work: Zero-Divisor Graph with Respect to an Ideal
cites work: Q4046212
cites work: An Ideal-Based Zero-Divisor Graph of a Commutative Ring
cites work: Der vollständige paare Graph auf nichtorientierbaren Flächen.
cites work: Q4773723
cites work: Q3637542
cites work: Zero-divisor graphs of genus one
cites work: Q5684700
cites work: Classification of Rings with Genus One Zero-Divisor Graphs
DOI: 10.1016/J.JALGEBRA.2008.09.040
description: scientific article; zbMATH DE number 5545838