non degenerate bilinear map for modules Let $R$ be a commutative ring with 1. Suppose $A,B$ are $R$-modules, $P:A\times B\to R$ is
a bilinear map that satisfies the following property: if $P(a,b)=0$ for all $b\in B$, then $a=0$. Then is the $R$-linear map $P':A\to B^*$, $P':a\mapsto P(a,b)$ an isomorphism? I could prove the injective part, but got stuck on proving $P'$ is surjective.
Plus, if I from these assumptions surjection is not necessarily true, what else should be assume about $P$ to ensure that $P'$ is surjective?

Hint: What you're talking about are the induced isomorphisms of a non-degenerate (not necessarily perfect) pairing. The definition of $P'$ is, however, a bit different. Here is the right one:
$$P': A \to B^*, a \mapsto P(a,-)$$
So the image of an $a \in A$ is really a dual map $B \to R$.
Are those modules finitely generated over $R$?
