TITLE: In what generality does the following isomorphism involving tensors and homs hold?
QUESTION [5 upvotes]: Let $R$ be a CRing and let $M,N$ be $R$-modules.  Let $M^*:=Hom_R(M,R)$.  I have seen the following isomorphism asserted in the case where $R$ is a field and $M$ and $N$ are f.g. vector spaces:
$M^*\otimes_R N\cong Hom_R(M,N)$.  
I can give a proof of this using a basis, but in what generality can we say this kind of thing?  Does it have a nice arrow-theoretic proof?
For some reason, I can't accept answers, so I apologize for not doing it if I am having trouble.  Maybe the mods can force my account to accept the answers? (To answer the gentleman's question, I am having javascript problems, so I fear that it won't help).

REPLY [5 votes]: Again ol' Bourbaki gives you an answer: when $M$ or $N$ are finitely projective modules, the canonical morphism $M^* \otimes N \longrightarrow \mathrm{Hom}(M,N)$ is an isomorphism (Algebra I, chapter II, 4.2, proposition 2).