TITLE: Finitely generated flat modules that are not projective
QUESTION [17 upvotes]: 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.

REPLY [6 votes]: Following Jack's gentle nudging, I'm posting the part of my comments not covered by rschwieb's nice post more prominently as an answer:
In general, this type of question can be answered quickly by consulting Lam's Lectures on Modules and Rings, Springer Graduate Texts, Vol. 189, whose numerous exercises are solved and put into further context by Lam himself in the accompanying text Exercises in Modules and Rings. The section §4E, Finitely generated flat modules of the Lectures contains some sufficient conditions on when f.g. flat modules are projective.
Here's exercise 4.17 with solution, giving Vasconcelos's example of a principal ideal $(a)$ in a commutative ring such that $R/(a)$ is flat but not projective:

This appears as Example 3.2 of Wolmer V. Vasconcelos, On finitely generated flat modules, Trans. Amer. Math. Soc. 138 (1969), 505-512.