Journal of Commutative Algebra

The clique ideal property

Thomas G. Lucas

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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

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

First available in Project Euclid: 16 December 2018

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Zero divisor graph clique maximum clique


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

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  • O.A. AbuGhneim, E.E. AbdAlJawad and H. Al-Ezeh, The clique number of $\Gamma(Z_{p^n}(\alpha))$, Rocky Mountain J. Math. 42 (2012), 1–14.
  • D.D. Anderson and M. Naseer, Beck's coloring of a commutative ring, J. Algebra 159 (1993), 500–514.
  • D.F. Anderson and P. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217 (1999), 434–447.
  • I. Beck, Coloring of commutative rings, J. Algebra 116 (1988), 208–226.
  • G.A. Cannon, K.A. Neuerburg and S.P. Redmond, Zero-divisor graphs of nearrings and semigroups, in Nearrings and nearfields, Springer, Dordrecht, 2005.
  • J. Huckaba, Commutative rings with zero divisors, Dekker, New York, 1988.
  • B. McDonald, Finite rings with identity, Pure Appl. Math. 28 (1974).
  • S.P. Redmond, On zero-divisor graphs of small finite commutative rings, Discr. Math. 307 (2007), 1155–1166.
  • ––––, Corrigendum to On zero-divisor graphs of small finite commutative rings, Discr. Math. 307 (2007), 2449–2452.