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In this paper we continue our study of annihilating-ideal graph of commutative rings, that was introduced in (The annihilating-ideal graph of commutative rings I, to appear in J. Algebra Appl.). Let R be a commutative ring with (R) be its set of ideals with nonzero annihilator and Z(R) its set of zero divisors. The annihilating-ideal graph of R is defined as the (undirected) graph (R) that its vertices are (R)* = (R)\{(0)} in which for every distinct vertices I and J, I — J is an edge if and only if IJ = (0). First, we study the diameter of (R). A complete characterization for the possible diameter is given exclusively in terms of the ideals of R when either R is a Noetherian ring or Z(R) is not an ideal of R. Next, we study coloring of annihilating-ideal graphs. Among other results, we characterize when either χ((R)) ≤ 2 or R is reduced and χ((R)) ≤ ∞. Also it is shown that for each reduced ring R, χ((R)) = cl ((R)). Moreover, if χ((R)) is finite, then R has a finite number of minimal primes, and if n is this number, then χ((R)) = cl ((R)) = n. Finally, we show that for a Noetherian ring R, cl ((R)) is finite if and only if for every ideal I of R with I 2  = (0), I has finite number of R-submodules.

THE ANNIHILATING-IDEAL GRAPH OF COMMUTATIVE RINGS II