Let $R$ be a Dedekind domain, $K$ its field of fractions, and $U$ a finite-dimensional vector space over $K$. Let $M$ be an $R$-submodule of $U$ that contains a basis of $U$ (so $M$ "spans" $U$).
Question: How does one produce an example of a module $M$ as described above that is not a free module? Or, how could one determine the existence of such a (non-free) module?
The setting comes from Cassels and Froelich's "Algebraic Number Theory", Chapter 1, Section 3. On the top half of page 10, the author refers to "remov[ing] the restriction that $M$ and $N$ are free modules", which suggests the existence of non-free $R$-modules embedded in a finite dimensional vector space over the field of fractions of a Dedekind domain $R$.
I suspect that this should not happen for the ring of integers of a number field, but have not tried to write up a proof of this. I have considered looking at an arbitrary Dedekind domain that is not a PID, and constructing $M$ using non-principal ideals somehow, but am not sure exactly how to go about this.
Is there some very simple example I'm not seeing? Or will finding such a module require looking much deeper into the structure of Dedekind domains?