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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 $Z{\left(R\right)}^{*}$ be its set of nonzero zero divisors. The zero divisor graph ${\Gamma }\left(R\right)$ of $R$ has vertices $Z{\left(R\right)}^{*}$ with two vertices $x$ and $y$ connected by an edge if and only if $xy=0$. D. F. Anderson and P. S. Livingston [J. Algebra 217, 434–447 (1999; Zbl 0941.05062)] showed that ${\Gamma }\left(R\right)$ is connected, has diameter $\text{diam}\left({\Gamma }\left(R\right)\right)\le 3$, and characterized the rings with $\text{diam}\left({\Gamma }\left(R\right)\right)\le 1$. M. Axtell, J. Coykendall and J. Stickles [Commun. Algebra 33, 2043–2050 (2005; Zbl 1088.13006)] investigated the zero divisor graphs of $R\left[X\right]$ and $R\left[\left[X\right]\right]$ and showed among other things that for $R$ noetherian not isomorphic to ${ℤ}_{2}×{ℤ}_{2}$, if one of ${\Gamma }\left(R\right)$, ${\Gamma }\left(R\left[X\right]\right)$, or ${\Gamma }\left(R\left[\left[X\right]\right]\right)$ 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 $1\le \text{diam}\left({\Gamma }\left(R\right)\right)\le \text{diam}\left({\Gamma }\left(R\left[X\right]\right)\right)\le \text{diam}\left({\Gamma }\left(R\left[\left[X\right]\right]\right)\right)\le 3$ and all possible sequences for these three diameters are given. For example, $\text{diam}\left({\Gamma }\left(R\right)\right)=\text{diam}\left({\Gamma }\left(R\left[X\right]\right)\right)=\text{diam}\left({\Gamma }\left(R\left[\left[X\right]\right]\right)\right)=3$ if and only if $R$ has more than two minimal primes and there is a pair of zero divisors $a$ and $b$ such that $\left(a,b\right)$ does not have nonzero annihilator. A similar characterization is given for $\text{diam}\left({\Gamma }\left(R\right)\right)$ and $\text{diam}\left({\Gamma }\left(R\left[X\right]\right)\right)$ when $R$ is not reduced.

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