Projectivity of factors of a nonzero finite free tensor product
= Projectivity of factors of a nonzero finite free tensor product
If $M\otimes_RN\cong R^n$ for a positive integer $n$, then both $M$ and $N$ are <projective modules>. Choose a finite expression for the inverse image of one basis vector. It produces a split surjection $N^r\to R$; tensoring the splitting with $M$ exhibits $M$ as a direct summand of the finite free module $(M\otimes_RN)^r$. Symmetry gives the result for $N$.