Over left noetherian rings and over semiperfect rings, every finitely generated flat module is projective. What are some examples of finitely generated flat modules that are not projective?
Compare to our question f.g. flat not free where all the answers are f.g. projective not free.
Best Answer
Over a von Neumann regular ring, every right module (and every left module) is flat. Let $V$ be an countable dimensional $F$ vector space, and let $R$ be the ring of endomorphisms of that vector space. It's known that $R$ is a von Neumann regular ring with exactly three ideals.
The nontrivial ideal $I$ consists of the endomorphisms with finite dimensional image. Then $R/I$ is flat but it cannot be projective. If it were projective, then $I$ would be a summand of $R$... but it is not, because it's an essential ideal.
A second example over any non-Artinian VNR ring: you can take $R/E$ for any maximal essential right ideal $E$ to get a nonprojective, simple flat module. The reasons are very much the same, since a proper essential right ideal can't be a summand of the ring.
You can even make a commutative version: take an infinite direct product of fields $\prod F_i$ (this is von Neumann regular). The ideal $I=\oplus F_i$ is an essential ideal, and $R/I$ is flat, nonprojective. (This one also has the added benefit of supplying examples of ideals which are projective but not free. Any summand of the ring will do, since the ring has IBN. The argument at the other post can be carried out again.)
Incidentally, Puninski and Rothmaler have written a nifty paper investigating which rings have all f.g. flat modules projective.