Journal of Commutative Algebra

The clique ideal property

Thomas G. Lucas

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Abstract

For a commutative ring $R$, one can form a graph $\Gamma (R)^*$ whose vertices are the zero divisors of $R$ (including $0$) and whose edges are the pairs $\{a,b\}$ where $ab=0$ with $a\ne b$. For this graph, a clique is a nonempty subset $X$ such that $ab=0$ for all $a\ne b$ in $X$. If $R$ is a finite ring, there is always a maximum clique of $\Gamma (R)^*$--a clique $X$ such that $|X|\ge |Y|$ for all cliques $Y$. We say that a finite ring $R$ has the clique ideal property if each maximum clique of $\Gamma (R)^*$ is an ideal of $R$. 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 $R=\mathbb Z_n[\scriptstyle{mathrm{X}} ]/(\scriptstyle{mathrm{X}} ^2)$ 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.

Article information

Source
J. Commut. Algebra, Volume 10, Number 4 (2018), 499-544.

Dates
First available in Project Euclid: 16 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.jca/1544950827

Digital Object Identifier
doi:10.1216/JCA-2018-10-4-499

Mathematical Reviews number (MathSciNet)
MR3892145

Zentralblatt MATH identifier
07003225

Subjects
Primary: 13M99: None of the above, but in this section
Secondary: 05C69: Dominating sets, independent sets, cliques

Keywords
Zero divisor graph clique maximum clique

Citation

Lucas, Thomas G. The clique ideal property. J. Commut. Algebra 10 (2018), no. 4, 499--544. doi:10.1216/JCA-2018-10-4-499. https://projecteuclid.org/euclid.jca/1544950827


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