Proving a module is projective.

Question

I read this statement and I have no idea how to prove it:

Every flat module over a PID is projective.

Is this even right?

Answer 1

This is false. For instance, over the PID $\mathbb{Z}$, the module $\mathbb{Q}$ is flat but not projective.

Answer 2

Eric Wofsey has already cut to the heart of the matter with the example, but here's some more information.

A classic theorem of Bass is that a ring is right perfect iff every left flat module is projective.

But a commutative, perfect domain has to be a field, since perfect rings satisfy the DCC on principal ideals (and nonfield domains do not.)

But every projective module over a PID is free. That's something that 'rhymes' with what you're saying.

Answer 3

This is not true, but over left noetherian rings and over semiperfect rings, every finitely generated flat module is projective.