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 $k$.

1. 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 $\operatorname{Res}_\varphi(M)$.

1. 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.