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Let R be a discrete valuation ring with fraction field F.
I believe it's straightforward to show that any torsion-free module M with the property that [itex]M \otimes_R F[/itex] is a finite dimensional F-vector space is of the form [itex]R^m \oplus F^n[/itex].
What if [itex]M \otimes_R F[/itex] is infinite dimensional?
I believe it's straightforward to show that any torsion-free module M with the property that [itex]M \otimes_R F[/itex] is a finite dimensional F-vector space is of the form [itex]R^m \oplus F^n[/itex].
What if [itex]M \otimes_R F[/itex] is infinite dimensional?