  We examine the dual of the so-called "hit problem", the latter being the
problem of determining a minimal generating set for the cohomology of products
of infinite projective spaces as module over the Steenrod Algebra $\mathcal{A}$
at the prime 2. The dual problem is to determine the set of $\mathcal
{A}$-annihilated elements in homology. The set of $\mathcal{A}$-annihilateds
has been shown by David Anick to be a free associative algebra. In this note we
prove that, for each $k \geq 0$, the set of {\it $k$ partially
$\mathcal{A}$-annihilateds}, the set of elements that are annihilated by $Sq^i$
for each $i\leq 2^k$, itself forms a free associative algebra.
