Let O be a Dedekind domain whose field of fractions K is
a global field. Let A be a finite-dimensional separable Kalgebra and let Λ be an O-order in A. Suppose that X is a
Λ-lattice such that K ⊗O X is free of some finite rank n over
A. Then X contains a (non-unique) free Λ-sublattice of rank
n. The main result of the present article ...
Let O be a Dedekind domain whose field of fractions K is
a global field. Let A be a finite-dimensional separable Kalgebra and let Λ be an O-order in A. Suppose that X is a
Λ-lattice such that K ⊗O X is free of some finite rank n over
A. Then X contains a (non-unique) free Λ-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 ]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