Suppose $M$ and $N$ are modules over a (commutative, unital) ring $S$. Let $R$ be a subring of $S$ such that $S,M$ and $N$ are all free, finitely generated modules over $R$.
Question: Under what conditions is $M\otimes_S N$ projective over $R$?
Of course, the tensor product over $R$, namely $M\otimes_R N$, is free over $R$ and $M \otimes_S N$ is the quotient of this by the $R$-submodule generated by $(sa)\otimes b-a\otimes (sb)$ where $a,b,s$ run over $R$-bases.
Note, that $S$ is only supposed to be module-free over $R$ not a free algebra over $R$!