This is a problem in "Foundations of Module and Ring Theory" of Wisbauer: " Let $R$ be a subring of the ring $S$ containing the unit of $S$. Show that a flat $R$-module $N$ is projective if and only if there is an exact sequence $0→K→P→N→0$ in the category of left $R$-modules with $P$ projective and $S⊗_RK$ finitely generated as an $S$-module."
It is a celebrated fact that under the hypotheses, if $M$ is a flat left $R$-module, then $S⊗_RM$ is a flat $S$- module. Now, the "only if" part is evidently true by choosing $P=N$ and $K=0$ with the identity map from $P$ to $N$. For the converse, I say that since $N$ is flat, so the above sequence is pure exact, hence it remains exact by tensoring with the $R$-module $S$. Also, since the middle term $P$ is a projective (hence flat) $R$-module, the flatness of $N$ yields that of $K$. Now, from this point on I could not proceed and demand some help. Thanks in advance!
A proof of your question can be found in this old paper.