Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 1109.13006
Lucas, Thomas G.
The diameter of a zero divisor graph. (English)
[J] J. Algebra 301, No. 1, 174-193 (2006). ISSN 0021-8693

Let $R$ be a commutative ring with $1$ and let $Z(R)^{\ast }$ be its set of nonzero zero divisors. The zero divisor graph $\Gamma (R)$ of $R $\ has vertices $Z(R)^{\ast }$ with two vertices $x$ and $y$ connected by an edge if and only if $xy=0$. \ {\it D. F.\ Anderson} and {\it P. S. Livingston} [J. Algebra 217, 434--447 (1999; Zbl 0941.05062)] showed that $\Gamma (R)$ is connected, has diameter $\text{diam}(\Gamma (R))\leq 3$, and characterized the rings with $\text{diam}(\Gamma (R))\leq 1$. {\it M. Axtell, J. Coykendall} and {\it J.\ Stickles} [Commun. Algebra 33, 2043--2050 (2005; Zbl 1088.13006)] investigated the zero divisor graphs of $R[X]$ and $ R[[X]]$ and showed among other things that for $R$ noetherian not isomorphic to $\Bbb{Z}_{2}\times \Bbb{Z}_{2}$, if one of $\Gamma (R)$, $\Gamma (R[X])$, or $\Gamma (R[[X]])$ has diameter $2$, then so do the other two. The main result of the paper under review is that for $R$ a reduced ring that is not an integral domain, we have $1\leq \text{diam}(\Gamma (R))\leq \text{diam}(\Gamma (R[X]))\leq \text{diam}(\Gamma (R[[X]]))\leq 3$ and all possible sequences for these three diameters are given. For example, $\text{diam}(\Gamma (R))=\text{diam}(\Gamma (R[X]))=\text{diam}(\Gamma (R[[X]]))=3$ if and only if $R$ has more than two minimal primes and there is a pair of zero divisors $a$ and $b$ such that $(a,b)$ does not have nonzero annihilator. A similar characterization is given for $\text{diam}(\Gamma (R))$ and $\text{diam}(\Gamma (R[X]))$ when $R$ is not reduced.
[Daniel D. Anderson (Iowa City)]

MSC 2000:: *13A99 General commutative ring theory
13F20 Polynomial rings
13F25 Formal power series rings
13A15 Ideals; multiplicative ideal theory
05C99 Graph theory

Citations: Zbl 0941.05062; Zbl 1088.13006

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]