tensor product and fixing component of a homomorphism

Let \(R\) be a ring, \(M,N,L\) be \(R\)-modules and

\begin{align*} f: M \otimes N \rightarrow L \end{align*}

a module-homomorphism for the tensor product

Then for arbitrary \(m \in M\), the map

\begin{align*} g: N \rightarrow& L \\ n \mapsto& f(m \otimes n) \end{align*}

is a module homomorphism

Let \(n_1,n_2 \in N\) and \(r \in R\), then by distributivity of an elementary tensor we get

\begin{align*} g(n_1 + r \cdot n_2) =& f(m \otimes (n_1 + r \cdot n_2)) \\ =& f((m \otimes n_1) + (m \otimes r \cdot n_2)) \\ =& f(m \otimes n_1) + f(m \otimes r \cdot n_2) \\ =& f(m \otimes n_1) + f(r \cdot (m \otimes n_2)) \\ =& f(m \otimes n_1) + r \cdot f(m \otimes n_2) \\ =& g(n_1) + r \cdot g(n_2) \end{align*}