Prove that map between modules is isomorphic. Let $R$ be a ring and $M$ and $R$-module. Let $M$ be finitely presented, i.e it is finitely generated with the surjective function $p: R^n \to M$ and $\ker(p)$ is also finitely generated.
Consider the right $R$-module $\text{Hom}_R(M,R)$ and for an $R$-module $N$ consider the map $$a:\text{Hom}_R(M,R) \otimes_R N \to \text{Hom}_R(M,N)$$ given by $f \otimes n \mapsto (m \mapsto f(m)n)$.
How can I prove that $a$ is an isomorphism if in additional to $M$ being finitely presented, I also know that it is projective?

Recall that a module is finitely generated projective iff it is finitely presented and flat. Thus here, projectivity of $M$ suffices. 
As $M$ is projective, $R^{n}=M \oplus ker(p)$. Now the isomorphism stated above by you holds in the case where $M$ is replaced by $R^{n}$. Restrict this isomorphism to $M$. 
Note - I am typing this on a phone, so could not write all the details. If you have any queries, please wait till I get home :)   
