This post is related to the following question: A counterexample to the isomorphism $M^{*} \otimes M \rightarrow Hom_R( M,M)$. I have been trying to isolate what hypothesis can be eliminated in these questions surrounding the tensor products of homomorphism groups and I was wondering if the following questions was formulated in such a way as to prevent a counterexample in the post quoted above.
Let $R$ be a commutative ring with identity. Let $M,N,P$ be $R$ modules and let $\theta : Hom_R (M,N) \otimes_R P \to Hom_R (M,N \otimes_R P)$ be the canonical mapping given by $(f,y) \mapsto (x \mapsto f(x \otimes y))$ for $f \in Hom_R (M,N)$ and $ y \in P$.
If M is a finitely generated projective module is the canonical mapping an isomorphism?