  In this paper we continue our study of annihilating-ideal graph of
commutative rings, that was introduced in Part I (see [5]). Let $R$ be a
commutative ring with ${\Bbb{A}}(R)$ 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 ${\Bbb{AG}}(R)$ that its vertices are
$\Bbb{A}(R)^* =\Bbb{A}(R)\hspace{-1mm}\setminus\{(0)\}$ in which for every
distinct vertices $I$ and $J$,
$I\hspace{-0.6mm}-\hspace{-1.7mm}-\hspace{-1.7mm}-\hspace{-0.5mm}J$ is an edge
if and only if $IJ=(0)$. First, we study the diameter of ${\Bbb{AG}}(R)$. 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 $\chi({\Bbb{AG}}(R))\leq 2$ or
$R$ is reduced and $\chi({\Bbb{AG}}(R))\leq \infty$. Also it is shown that for
each reduced ring $R$, $\chi(\Bbb{AG}(R))= cl(\Bbb{AG}(R))$. Moreover, if
  $\chi(\Bbb{AG}(R))$ is finite, then $R$ has a finite number of minimal
primes, and if $n$ is this number, then $\chi(\Bbb{AG}(R))= cl(\Bbb{AG}(R))=
n$. Finally, we show that for a Noetherian ring $R$, $cl(\Bbb{AG}(R))$ is
finite if and only if for every ideal $I$ of $R$ with $I^2=(0)$, $I$ has finite
number of $R$-submodules.

Commutative Algebra;Rings and Algebras