Let R be a commutative ring, with ๐ธ(R) its set of ideals with nonzero annihilator. In this paper and its sequel, we introduce and investigate the annihilating-ideal graph of R, denoted by ๐ธ๐พ(R). It is the (undirected) graph with vertices ๐ธ(R)* โ ๐ธ(R)\{(0)}, and two distinct vertices I and J are adjacent if and only if IJ = (0). First, we study some finiteness conditions of ๐ธ๐พ(R). For instance, it is shown that if R is not a domain, then ๐ธ๐พ(R) has ascending chain condition (respectively, descending chain condition) on vertices if and only if R is Noetherian (respectively, Artinian). Moreover, the set of vertices of ๐ธ๐พ(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, ๐ธ๐พ(R) has n vertices (n โฅ 1) if and only if R has only n nonzero proper ideals. Next, we study the connectivity of ๐ธ๐พ(R). It is shown that ๐ธ๐พ(R) is a connected graph and diam (๐ธ๐พ)(R) โค 3 and if ๐ธ๐พ(R) contains a cycle, then gr (๐ธ๐พ(R)) โค 4. Also, rings R for which the graph ๐ธ๐พ(R) is complete or star, are characterized, as well as rings R for which every vertex of ๐ธ๐พ(R) is a prime (or maximal) ideal. In Part II we shall study the diameter and coloring of annihilating-ideal graphs.