  It's well known that the functional Hilbert space over the unit ball in
$B_{d} \in C^d$, with kernel function $K(z,w)=\frac{1}{1-z_{1}w_{1}-...
-z_{d}w_{d}}$, admits a natural $A(B_{d})$-module structure. We show the rank
of a nonzero submodule is infinity if and only if the submodule is of infinite
codimension. Together with Arveson's dilation theory, our result shows that
Hilbert modules stand in stark contrast with Hilbert basis theorem for
algebraic modules. This result answers a question of Arveson.
