If over a commutative ring $R$ we have that $M\otimes N=R^n$, $n\neq 0$, need we have that $M$ and $N$ are finitely generated projective?
We have finite generation, because if $M\otimes N$ is generated by $\sum_i a_{ij} x_i^j\otimes y_i^j$, then we have $x_{i_1}^{j_1}\otimes y_{i_2}^{j_2}\otimes x^{j_3}_{i_3}$ generating $M^n$. Projecting onto $M$, we see that $x_i^j$ generate $M$. My question is can we show that this map splits?
I am able to verify if in the case that $n=1$. I can't personally think of an example where it is not true, but I am having problems constructing a splitting. Any other solution is appreciated too.
