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I am not sure this is standard terminology but the notes I am using has a mapping defined as follows. In particular I need help with how to write out the definition in some related exercises.

Let $R$ be a commutative ring with identity. Let $M,N,P$ be $R$ modules and let $\theta : Hom_A (M,N) \otimes_R P \rightarrow Hom_A (M,N \otimes_R P)$ be the canonical mapping.

Is the canonical mapping theta given by $(f,y) \rightarrow (x \rightarrow (x \rightarrow f(x \otimes y))$ for $f \in Hom_A (M,N)$ and $ y \in P$?

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    I think you mean the linear map induced by the bilinear map $(f,y) \mapsto (x \mapsto f(x) \otimes y)$. (See Mariano's answer.)2011-09-30

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There is a map $\theta : Hom_A (M,N) \otimes_R P \rightarrow Hom_A (M,N \otimes_R P)$ such that $\theta(f\otimes p)(m)=f(m)\otimes p$ for all $f\in Hom_A(M,N)$, all $p\in P$ and all $m\in M$. This is the map one usually refers to as the canonical map in this context.

It makes for a good exercise to check that there is indeed such a map, by the way!