TITLE: Flat Module and Torsion-Free Module 
QUESTION [27 upvotes]: All rings in this question are integral.
It is known that flat modules are torsion-free. Conversely, torsion-free modules over Prüfer domain (in particular, Dedekind domain) are flat, please see here. My questions are:


Is there a general condition under which a torsion free module is flat?
What is the simplest example of torsion-free non-flat module?


edit: Since the example in Georges's answer is the coordinate ring of a singular curve, I want to ask the same questions for the coordinate ring $R$ of a smooth algebraic variety: 


Is there a general condition under which a torsion free module over $R$ is flat? 
what is the simplest example of torsion-free non-flat module over  $R$

REPLY [5 votes]: If the question is only for commutative domains, then yes, torsion-free=flat iff it is Pruefer (see e.g. Fuchs/Salce, Thm. VI.9.10, actually due to Warfield), as was mentioned earlier. If not only domains are in question, then it can be said that Warfield showed that it is true whenever the ring's localizations at maximal ideals are valuation rings (let's call them Pruefer rings). I haven't checked if those are the only commutative rings for which this is true. If one leaves the realm of commutative rings, there is a much bigger class, so-called RD-rings, for which it is true. These are the rings for which Relative Divisibility is the same as purity (just as over the integers). An example is the first Weyl algebra over any field of char 0 (this is even a domain, though not commutative, so possibly not of interest to the poser of the question).