Let $A$ be an algebra (possibly non-commutative). Let $M,N$ be $A-A$-bimodules. Suppose that $M\otimes_AN\cong A$. Can we conclude that $M$ is left $A$-projective?
I tried many things but ultimately failed to prove this. It is worth noting though that $N$ should inject into the dual of $M$ (I think) and that $(N\otimes_A M)^{\otimes 2}\cong N\otimes_A M$. These things look useful.