This is a question out of Donald Passman's "A Course in Ring Theory".
Let $R$ be a Dedekind domain with field of fractions $F$, and let $V$ be a finitely generated $R$ submodule of $F^n:=F\oplus\dots\oplus F$. Passman asks to show whether $V$ is projective.
Now I have already proven that over a semihereditary ring any finitely generated submodule of a free module is projective. Obviously $R$ is semihereditary, being a Dedekind domain, but I am struggling to see whether $F$ is a free $R$ module. If it is then my proof is done, but if not I'll need another approach. So if anyone can provide hints on how to prove that $F$ is a free $R$ module it would be much appreciated, if it is not then a push in the right direction would be very useful.
EDIT: As proven here, there is a counterexample to $F$ being free, which means that my proposed strategy won't work. Unfortunately I cannot think of another way to proceed, so any prod in the right direction will be much appreciated.
Donald Passman. A Course in Ring Theory. Wadsworth and Brooks/Cole, 1st edition, 1991.