  Let $R$ be a commutative ring with ${\Bbb{A}}(R)$ its set of ideals with
nonzero annihilator. In this paper and its sequel, we introduce and investigate
the {\it annihilating-ideal graph} of $R$, denoted by ${\Bbb{AG}}(R)$. It is
the (undirected) graph with vertices
${\Bbb{A}}(R)^*:={\Bbb{A}}(R)\setminus\{(0)\}$, and two distinct vertices $I$
and $J$ are adjacent if and only if $IJ=(0)$. First, we study some finiteness
conditions of ${\Bbb{AG}}(R)$. For instance, it is shown that if $R$ is not a
domain, then ${\Bbb{AG}}(R)$ has ACC (resp., DCC) on vertices if and only if
$R$ is Noetherian (resp., Artinian). Moreover, the set of vertices of
${\Bbb{AG}}(R)$ and the set of nonzero proper ideals of $R$ have the same
cardinality when $R$ is either an Artinian or a decomposable ring. This yields
for a ring $R$, ${\Bbb{AG}}(R)$ has $n$ vertices $(n\geq 1)$ if and only if $R$
has only $n$ nonzero proper ideals. Next, we study the connectivity of
${\Bbb{AG}}(R)$. It is shown that ${\Bbb{AG}}(R)$ is a connected graph and
$diam(\Bbb{AG})(R)\leq 3$ and if ${\Bbb{AG}}(R)$ contains a cycle, then
$gr({\Bbb{AG}}(R))\leq 4$. Also, rings $R$ for which the graph ${\Bbb{AG}}(R)$
is complete or star, are characterized, as well as rings $R$ for which every
vertex of ${\Bbb{AG}}(R)$ is a prime (or maximal) ideal. In Part II we shall
study the diameter and coloring of annihilating-ideal graphs.
