The clique ideal property
type: Item
instance of: scholarly article
title: The clique ideal property
zbMATH Open document ID: 1408.13067
DOI: 10.1216/JCA-2018-10-4-499
author: Thomas G. Lucas
published in: Journal of Commutative Algebra
publication date: 2019-01-18T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.jca/1544950827
review text: Many papers appear on the topic of graphs from commutative rings. The first ever graph construction from a commutative ring is the zero-divisor graph by \textit{I. Beck} [J. Algebra 116, No. 1, 208--226 (1988; Zbl 0654.13001)]. Afterwards, it was modified by \textit{D. F. Anderson} and \textit{P. S. Livingston} [J. Algebra 217, No. 2, 434--447 (1999; Zbl 0941.05062)]. This is a paper on the graph defined by Beck. The modified zero-divisor graph is well studied on various perspections of rings and graphs. For a commutative ring \(R\), one can form a graph \(\Gamma(R)^*\) (as defined by Beck) whose vertices are the zero divisors of \(R\) (including 0) and whose edges are the pairs \(\{a, b\}\) where \(ab = 0\) with \(a\neq b\). For this graph, a clique is a nonempty subset \(X\) such that \(ab = 0\) for all \(a\neq b\) in \(X\). If \(R\) is a finite ring, there is always a maximum clique of \(\Gamma(R)^*\) -- a clique \(X\) such that \(|X| = |Y|\) for all cliques \(Y\). A finite ring \(R\) has the clique ideal property if each maximum clique of \(\Gamma(R)^*\) is an ideal of \(R\). This paper deals with rings with clique ideal property of commutative rings. If \(R = S\oplus T\) where both \(S\) and \(T\) are finite rings with the clique ideal property and neither \(S\) nor \(T\) is a field, then \(R\) has the clique ideal property. The converse does not hold. For each positive integer \(n > 1,\) the ring \(R = \mathbb{Z}_n[x]/(x^2)\) is a finite ring with the clique ideal property. In contrast, \(\mathbb{Z}_n\) has the clique ideal property if and only if \(n\) is either a prime or a perfect square.
reviewed by: T. Tamizh Chelvam
Mathematics Subject Classification ID: 13M99
Mathematics Subject Classification ID: 05C69
zbMATH DE Number: 7003225
zbMATH Keywords: zero divisor graph
zbMATH Keywords: clique
zbMATH Keywords: maximum clique
MaRDI profile type: Publication
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cites work: On zero-divisor graphs of small finite commutative rings
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OpenAlex ID: W2904212573
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description: scientific article; zbMATH DE number 7003225