  Let $R$ be a unital ring satisfying the invariant basis number property, that
every stably free $R$-module is free, and that the complex of partial bases of
every finite rank free module is Cohen--Macaulay. This class of rings includes
every ring of stable rank $1$ (e.g. any local, semi-local or Artinian ring),
every Euclidean domain, and every Dedekind domain $\mathcal{O}_S$ of arithmetic
type where $|S| > 1$ and $S$ contains at least one non-complex place. Extending
recent work of Galatius--Kupers--Randal-Williams and Kupers--Miller--Patzt, we
prove that the sequence of general linear groups $\operatorname{GL}_n(R)$
satisfies slope-$1$ homological stability with $\mathbb{Z}[1/2]$-coefficients.
