  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 an explicit bound that is 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.

Number Theory;Rings and Algebras