tensoring to change the ring as functor
1. Proposition
Let \(R,S\) be commutative (rings), \(\varphi: R \rightarrow S\) a ring-homomorphism and \(M\) an \(R\)-module. Then tensor is a functor from \(\mathrm{SMod}\) to \(\mathrm{RMod}\) (cf. category RMod)
\begin{align*} (R \otimes_{S} -): \mathrm{SMod} \rightarrow& \mathrm{RMod} \\ M \mapsto& (R \otimes_S M) \\ (f: M \rightarrow M') \mapsto& f': (R \otimes_S M) \rightarrow (R \otimes_S M') \end{align*}where \(f'\) is defined by
\begin{align*} f'(r \otimes m) = r \otimes f(m) \end{align*}see: elementary tensor
2. Proof
2.1. ring homomorphism
Let \(r_1,r_2 \in R\) and \(m \in M\), then
\begin{align*} r_1 \cdot_R f'(r_2 \otimes m) =& r_1 \cdot_R (r_2 \otimes f(m)) \\ =& (r_1 \cdot r_2 \end{align*}