  Let $\mathcal{O}$ be a Dedekind domain whose field of fractions $K$ is a
global field. Let $A$ be a finite-dimensional separable $K$-algebra and let
$\Lambda$ be an $\mathcal{O}$-order in $A$. Let $n$ be a positive integer and
suppose that $X$ is a $\Lambda$-lattice such that $K \otimes_{\mathcal{O}} X$
is free of rank $n$ over $A$. Then $X$ contains a (non-unique) free
$\Lambda$-sublattice of rank $n$. The main result of the present article is to
show there exists such a sublattice $Y$ such that the generalised module index
$[X : Y]_{\mathcal{O}}$ has explicit upper bounds with respect to division that
are independent of $X$ and can be chosen to satisfy certain conditions. We give
examples of applications to the approximation of normal integral bases and
strong Minkowski units, and to the Galois module structure of rational points
over abelian varieties.
