Restriction of scalars

Let $R$ and $S$ be commutative rings , and let $\varphi: R \to S$ be a ring homomorphism. If $M$ is an $S$-module, the restriction of scalars of $M$ along $\varphi$ is the $R$-module

defined as follows:

• As an abelian group, $\operatorname{Res}_\varphi(M)$ is the same underlying abelian group as $M$.
• The $R$-action is given by $r\cdot m := \varphi(r)m \quad \text{for } r\in R,\ m\in M,$ where the multiplication on the right is the original $S$-module structure on $M$.

This construction is functorial in $M$ and defines a forgetful functor

It is faithful and exact (it does not change the underlying abelian groups or group homomorphisms).

Restriction of scalars is the natural companion to extension of scalars , and in many settings these form an adjoint pair (extension is left adjoint to restriction).

Examples

1. From a polynomial algebra to the base field.
   Let $k$ be a field and $\varphi: k \hookrightarrow k[x]$ the usual inclusion. If $M = k[x]$ is viewed as a $k[x]$-module over itself, then $\operatorname{Res}_\varphi(M)$ is just the underlying $k$-vector space of polynomials, which is infinite-dimensional over $k$.

2. Forgetting an $S$-module to a $\mathbb{Z}$-module.
   For the canonical surjection $\varphi: \mathbb{Z}\to \mathbb{Z}/n\mathbb{Z}$, any $\mathbb{Z}/n\mathbb{Z}$-module $M$ becomes a $\mathbb{Z}$-module by restriction. Concretely, $\operatorname{Res}_\varphi(M)$ is the underlying abelian group of $M$, and it satisfies $nM=0$, i.e. $n$ lies in the annihilator of $\operatorname{Res}_\varphi(M)$.

3. Restriction along localization.
   If $S=R_f$ is the localization of $R$ at a single element $f\in R$, then every $R_f$-module restricts to an $R$-module along $R\to R_f$. For instance, $R_f$ itself (as an $R_f$-module) becomes an $R$-module in which multiplication by $f$ is invertible; compare this with localization of modules .