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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) * be its set of nonzero zero divisors. The zero divisor graph Γ(R) of R has vertices Z(R) * 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 Γ(R) is connected, has diameter diam(Γ(R))3, and characterized the rings with diam(Γ(R))1. M. Axtell, J. Coykendall and 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 2 × 2 , if one of Γ(R), Γ(R[X]), or Γ(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 1diam(Γ(R))diam(Γ(R[X]))diam(Γ(R[[X]]))3 and all possible sequences for these three diameters are given. For example, diam(Γ(R))=diam(Γ(R[X]))=diam(Γ(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 diam(Γ(R)) and diam(Γ(R[X])) when R is not reduced.

13A99General commutative ring theory
13F20Polynomial rings and ideals
13F25Formal power series rings
13A15Ideals; multiplicative ideal theory
05C99Graph theory