Let $M$ and $N$ be a finitely generated projective module over a ring $R$. Suppose we have a non degenerate bilinear pairing $\langle \ \cdot \ ,\ \cdot\ \rangle: M \times N \to R$.
I want to show $M$ is isomorphic to the dual $N^*$ of $N$.
The injectivity of $M$ into $N^*$ follows from the non degeneracy of the pairing by definiting a map $x \mapsto \langle x, \cdot\rangle$. What I cannot prove is surjectivity.
If I impose a condition that $R$ is an injective module then I think surjectivity also follows. But I want to prove it without any condition on $R$
Any help is appriciated.