Let $F/\mathbb Q$ be a field extension, let $\mathfrak o$ be an order in $F$ (this is, $\mathfrak o$ is a subring which is free as a $\mathbb Z$-module, of rank $[F:\mathbb Q]$).
Are there any results about the classification of finitely generated torsion-free modules over $\mathfrak o$?
I know that, if $\mathfrak o$ is the ring of integers of $F$, it is a Dedekind domain, and in particular any finitely generated, torsion-free module is projective, and can be written as a free module times a fractional ideal, so that case is solved.
Maybe my question can be split in two, namely:
Is any finitely generated torsion free module over $\mathfrak o$ projective?
Trying to classify them
To answer the first question: no, if you have a domain such that all ideals are projective (this property is also known as hereditary), then it is a Dedekind domain. Now $\mathfrak{o}$ is Noetherian, so if all f.g. torsion free modules are projective, then in particular all ideals are projective and thus $\mathfrak{o}$ is a Dedekind domain. This is not true unless $\mathfrak{o}=\mathcal O_F$.