  Let $A$ be a finitary algebra over a finite field $k$, and $A$-$mod$ the
category of finite dimensional left $A$-modules. Let $\mathcal{H}(A)$ be the
corresponding Hall algebra, and for a positive integer $r$ let $D_{r}(A)$ be
the subspace of $\mathcal{H}(A)$ which has a basis consisting of isomorphism
classes of modules in $A$-$mod$ with at least $r+1$ indecomposable direct
summands. If $A$ is hereditary of type $A_{n}$, then $D_{r}(A)$ is known to be
the kernel of the map from the twisted Hall algebra to the quantized Schur
algebra indexed by $n+1$ and $r$. For any $A$, we determine necessary and
sufficient conditions for $D_{r}(A)$ to be an ideal and some conditions for
$D_{r}(A)$ to be a subring of $\mathcal{H}(A)$. For $A$ the path algebra of a
quiver, we also determine necessary and sufficient conditions for $D_{r}(A)$ to
be a subring of $\mathcal{H}(A)$.
