Zero-divisor graphs of genus one
type: Item
instance of: scholarly article
title: Zero-divisor graphs of genus one
zbMATH Open document ID: 1106.13029
author: Hsin-Ju Wang
published in: Journal of Algebra
publication date: 2006-12-07T00:00:00Z
review text: The goal of the present paper is to describe all rings of two special types which have genus at most one. The first type studied is represented by finite direct products of \(\mathbb{Z}_{p_i^{n_i}}\) (where all \(p_i\) are primes), while the second by the factor rings \(\mathbb{Z}_n[x]/(x^m)\) (where \(m\) and \(n\) are at least \(2\)). Moreover, Theorem 1.4 of \textit{S. Akbari, H. R. Maimani} and \textit{S. Yassemi} [J. Algebra 270, No. 1, 169--180 (2003; Zbl 1032.13014)] is generalized (see Theorem 3.6) and there is given an affirmative answer to a question raised in [loc. cit.] (``Is it true that, for any local ring \(R\) of cardinality 32, which is not a field, the graph \(U_{393}(R)\) introduced by \textit{D. F. Anderson} and \textit{P. Livingston} [J. Algebra 217, No. 2, 434--447 (1999; Zbl 0941.05062)] is not planar?'').
Mathematics Subject Classification ID: 13M05
zbMATH DE Number: 5077817
reviewed by: Marius Tărnăuceanu
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2006.01.057
OpenAlex ID: W2047344301
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: CYCLES AND SYMMETRIES OF ZERO-DIVISORS
cites work: SOLUTION OF THE HEAWOOD MAP-COLORING PROBLEM
DOI: 10.1016/J.JALGEBRA.2006.01.057
description: scientific article; zbMATH DE number 5077817