Classification of rings with projective zero-divisor graphs
type: Item
instance of: scholarly article
title: Classification of rings with projective zero-divisor graphs
zbMATH Open document ID: 1143.13029
author: Hung-Jen Chiang-Hsieh
published in: Journal of Algebra
publication date: 2008-05-21T00:00:00Z
review text: Let \(R\) be a commutative ring with identity, \(\Gamma(R)\) the zero-divisor graph of \(R.\) The paper investigates the embedding of \(\Gamma(R)\) into a non-orientable compact surface. If \(S\) is such a surface, \(S\) can be written as a connected sum of finite copies of projective planes. The number of these copies is called the crosscap number of \(S.\) A non-planar graph is called projective if it can be embedded into the projective plane. In the paper, the crosscap number of surfaces in which \(\Gamma(R)\) can be embedded is investigated, as well as all finite rings with projective zero-divisor graphs.
reviewed by: Cristodor-Paul Ionescu
Mathematics Subject Classification ID: 13M05
Mathematics Subject Classification ID: 05C99
zbMATH DE Number: 5277613
zbMATH Keywords: projective zero-divisor graph
zbMATH Keywords: crosscap number
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2007.10.015
OpenAlex ID: W2002809788
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: Zero-divisor graphs, von Neumann regular rings, and Boolean algebras.
cites work: The zero-divisor graph of a commutative ring
cites work: Coloring of commutative rings
cites work: Blocks and the nonorientable genus of graphs
cites work: Orientable and non orientable genus of the complete bipartite graph
cites work: Q3083011
cites work: Q3757929
cites work: Q5534009
cites work: CYCLES AND SYMMETRIES OF ZERO-DIVISORS
cites work: Q4773723
cites work: Q3637542
cites work: Infinite Planar Zero-Divisor Graphs
cites work: Q3847965
cites work: Zero-divisor graphs of genus one
cites work: Classification of Rings with Genus One Zero-Divisor Graphs
DOI: 10.1016/J.JALGEBRA.2007.10.015
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description: scientific article; zbMATH DE number 5277613