For an $R$-module $M$, the dual is given by $M^*=Hom(M,R)$. I was wondering, can $M$ always be written in the form $M=N^*$ for some $R$-module $N$? If not, are there any known conditions for it?
I know that for some special cases, it is possible. For example, the reflexive modules. Also, since $(\bigoplus_i N_i)^*=\prod_i N_i^*$, this "class" of modules is closed under (possibly infinite) products (and limits in general).
The reason I want to know it is because if it is true for modules on rings, it is also true on quasi-coherent modules of schemes, and this would be a proof that all quasi-coherent modules are representable by schemes.
EDIT: It is indeed not true in general; and in cases when it is true, the morphism $ev: N\to N^{**}$ must have a left inverse (so in particular, it is injective). But that isn't a sufficient condition either, as the first commenter gave a counterexample.
For finitely generated modules over Noetherian integral domains, being the dual of another finitely generated module is equivalent to being reflexive; see e.g. Thm. 2.8 of Karl Schwede's notes.