# Question about the definition of a canonical mapping of tensor products

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 $y \in P$?

• The rule you have given does not make sense for a map $Hom_A (M,N) \otimes_R P \rightarrow Hom_A (M,N \otimes_R P)$ – Mariano Suárez-Álvarez Sep 30 '11 at 17:28
• Thanks for your response. Is there some well defined map that is similar to the one above that you are aware of could be a possible candidate in this situation? – user7980 Sep 30 '11 at 17:35
• Dear user7980: I think there is a typo. Please check the display. Also you should try to define a linear map from a tensor product by a certain bilinear map. – Pierre-Yves Gaillard Sep 30 '11 at 17:39
• Thanks for the help. I think the problem was the map was missing a parenthesis at $f(x \otimes y)$. – user7980 Sep 30 '11 at 17:45
• 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.) – Pierre-Yves Gaillard Sep 30 '11 at 17:49

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.