restriction of scalars functor

Let \(R,S\) be rings, \(\varphi: R \rightarrow S\) a ring-homomorphism. Then the restriction of scalars functor is defined as functor from category SMod to category RMod

\begin{align*} \varphi^{*}: \mathrm{SMod} \rightarrow& \mathrm{RMod} \\ M \mapsto& M \\ (f: M \rightarrow N) \mapsto& (f: M \rightarrow N) \end{align*}

see:

• restriction of scalars and module homomorphisms
• restriction of scalars of a module for a ringhomomorphism

functorality follows from the construction