TITLE: Example of a finitely generated flat module which is not free
QUESTION [6 upvotes]: I couldn't come up with an example of a finitely generated flat module which is not free. I know that over local rings, freeness and flatness are equivalent. So the ring cannot be a local ring.

REPLY [3 votes]: Rings over which every module is flat are called absolutely flat, or Von Neumann regular. If $I$ is any nontrivial ideal of an absolutely flat ring, then $R/I$ is finitely generated flat over $R$, but not free (since $R/I \neq 0$ and $\mathrm{Ann}(R/I)=I \neq 0$). Products of fields are absolutely flat.