Let $A$ be a nuclear Frechet algebra with unit. Let $M$ be a right Frechet $A$-module and $N$ be a left Frechet $A$-module. Both $M$ and $N$ are assumed to be non-degenerate. We can define the projective tensor product $M\hat{\otimes}_A N$ as in The Homology of Banach and Topological Algebras Chapter II, $4.1. Actually we have another, inductive tensor product, but these two construction coincide for Frechet modules.
On the other hand, we can forget the topologies and consider the algebraic tensor product $M\otimes_A N$. In general $M\hat{\otimes}_A N\not\cong M\otimes_A N$. Nevertheless, if both $M$ and $N$ are finitely generated projective $A$-modules, which means they are direct summands of finite direct sums of $A$, then it is clear that $M\hat{\otimes}_A N\cong M\otimes_A N$.
My question is: if we only assume $M$ is finitely generated and projective, do we still have $M\hat{\otimes}_A N\cong M\otimes_A N$? If not, do we have a counter-example?
