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The diameter of a zero divisor graph. (English) Zbl 1109.13006
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.

MSC:
13A99General commutative ring theory
13F20Polynomial rings and ideals
13F25Formal power series rings
13A15Ideals; multiplicative ideal theory
05C99Graph theory
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Full Text: DOI
References:
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[2] Axtell, M.; Coykendall, J.; Stickles, J.: Zero-divisor graphs of polynomials and power series over commutative rings. Comm. algebra 6, 2043-2050 (2005) · Zbl 1088.13006
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