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While thinking about this question (and specifically YCor's remarks), I tried to remember what can be said about rings such that every torsion-free module is free, and I could not; and such things, while certainly well-known, are very difficult to search for (either using Google or in books). We need a summary.
So let me ask the following:

Over any (commutative) ring, a free module is projective, a projective module is flat and a flat module is torsion-free. For each of these implications, what conditions are known on the ring such that the converse holds? In other words:

  • Over which rings is every torsion-free module flat?

  • Over which rings is every flat module projective?

  • Over which rings is every projective module free?

In each case, the class of rings in question may have a name, and there are a number of standard necessary or sufficient, or maybe necessary-and-sufficient conditions, for the implication to hold, so I am asking for a summary and references. Maybe add the same questions when restricted to finitely generated modules. And for bonus points, remove the standing “commutative” assumption.

I am aware of many partial answers to this question, like the fact that torsion-free modules over Prüfer domains are flat, that finitely generated flat modules over a Noetherian or local ring are projective, that rings over which every flat module is projective are known as “perfect” (but I know essentially nothing about them), and that projective modules over local rings or finitely generated projective modules over PIDs are free. The point of this question would be to gather all these kinds of facts in a single place, if possible in a readable way. (Of course, if a book already exists that does this, please refer to it!)

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    $\begingroup$ This should probably be community wiki, but I didn't remember how this works: was I supposed to mark the question as such? (I didn't see any checkbox.) Or do I need to ask a moderator to do it? $\endgroup$
    – Gro-Tsen
    Commented May 6, 2021 at 13:03
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    $\begingroup$ The last class of rings goes by the name, happily enough, "projective-free". Cohn's books mention a few classes of (commutative and noncommutative) rings with this property. For the second question you might start with the paper "When is a flat module projective" and search the relevant literature. $\endgroup$
    – Pace Nielsen
    Commented May 6, 2021 at 15:10
  • $\begingroup$ Let me assume that the rings are commutative and Noetherian. Since torsion-freeness is nicer for domains, let me also assume domain. Further, if the ring is a field, all of the above are true, let me also assume non-field. Then, the first question means the ring must be a Dedekind domain. No such rings (only fields) for the second. Third, there are probably many (like polynomial rings over a field?). $\endgroup$
    – Mohan
    Commented May 6, 2021 at 21:38
  • $\begingroup$ A commutative ring has every flat module projective iff $A = \prod A_i$ where each $A_i$ has exactly one prime ideal $P_i$ and $P_i$ is $T$-nilpotent. $\endgroup$
    – Badam Baplan
    Commented May 8, 2021 at 19:30
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    $\begingroup$ The rings (not necessarily commutative) for which every flat module is projective are called perfect rings. For commutative rings, it means semi-local + some nilpotence condition on the radical. $\endgroup$
    – abx
    Commented May 18, 2021 at 16:48

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