---- > [!definition] Definition. ([[restriction of scalars]]) > Let $R,S$ be [[commutative ring|commutative]] [[ring|rings]]. > Let $f:R \to S$ be a [[ring homomorphism]], and let $N$ be an $S$-[[module]] with structure [[ring homomorphism|homomorphism]] $\sigma :S \to \text{End}_{\mathsf{Ab}}(N).$ Composing with $f$, $f \circ \sigma : R \to \text{End}_{\mathsf{Ab}}(N)$ defines an action *of $R$* on the [[abelian group]] $N$, and hence an $R$-[[module]] structure on $N$. Explicitly, multiplication looks like $rn:=f(r)n.$ Since $S$ is [[commutative ring|commutative]], this defines in fact an $(R,S)$-[[bimodule]] structure of $N$. Furthermore, $S$-linear [[linear map|homomorphisms]] are in fact $R$-linear; this assignment is (covariantly) [[covariant functor|functorial]] $f_{*}:R$-$\mathsf{Mod} \to S$-$\mathsf{Mod}$. This functor is called **restriction of scalars**. ^definition > [!note] Terminology. > If $f: R\hookrightarrow S$ is [[injection|injective]], so that $R$ may be viewed as a [[subring]] of $S$, then all we are doing is viewing $N$ as a [[module]] on a 'restricted' range of scalars, hence the terminology. ^note ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```