  Let $\{A_m\}$ be a pro system of associative commutative, not necessarily
unital, rings. Assume that the pro systems $\{\mathrm{Tor}^{\mathbb{Z}\ltimes
A_m}_i(\mathbb{Z},\mathbb{Z})\}_m$ vanish for all $i>0$. Then we prove that the
sequence \[
  \{H_l(\mathrm{GL}_n(A_m))\}_m \to \{H_l(\mathrm{GL}_{n+1}(A_m))\}_m \to
\{H_l(\mathrm{GL}_{n+2}(A_m)\}_m \to \cdots \] stabilizes up to pro
isomorphisms for $n$ large enough than $l$ and the stable range of $A_m$'s.
