Let O be an R-order for some Dedekind domain R, let F be the field of fractions of R and D be a division algebra over F. A fractional left ideal of O is an R-lattice I in D such that OI in I (I absorbs multiplication of O on the left).
How do I prove that in such a division algebra, every nonzero fractional (left) ideal of O is a complete R-lattice?