Let $R$ be a (not necessarily commutative) ring, $M$ a left $R$-module, and $N$ a right $R$-module. We say that a pairing 
$$
\langle -,-\rangle:M \otimes_R N \to R
$$
is non-degenerate if, for all $n \in N$ there exists an $m \in M$ such that $\langle m,n\rangle \neq 0$, \textbf{and} for all $m \in M$, there exists an $n \in N$ such that $\langle m,n\rangle \neq 0$.

Such a pairing will give embeddings
$$
M \hookrightarrow N^*,  ~~~~~ N \hookrightarrow M^*,
$$
where $M^*$ and $N^*$ denote the dual modules of $M$ and $N$ respectively. In general (even for infinite-dimensional vector spaces) this will not give isomorphisms
$$
N \simeq M^*, ~~~~ M \simeq N^*.
$$
However, if we assume that $M$ and $N$ are finitely-generalted projective, then does non-degeneracy imply that we get isomorphisms? 

If it fails in the general noncommutative setting, I would still be interested in a positive answer in the commutative setting.