Restriction of scalars

Definition (restriction of scalars)

Let $\varphi: R \to S$ be a ring homomorphism (typically with $R,S$ commutative and unital).
If $M$ is an $S$-module , then restriction of scalars along $\varphi$ equips $M$ with an $R$-module structure by

where the multiplication on the right is the given $S$-module action.

This construction is often denoted $\mathrm{Res}^{S}_{R}(M)$, and it defines a functor

Relationship to extension of scalars

Restriction of scalars is the “forgetful” direction opposite to extension of scalars (which typically uses the tensor product).

Examples

1. Field extension (vector spaces).
   If $k \subset K$ is a field extension and $V$ is a $K$-vector space, then by restricting scalars along $k \hookrightarrow K$, the same underlying abelian group becomes a $k$-vector space (often of larger dimension).

2. Quotient map (annihilating an ideal).
   Let $\pi: R \to R/I$ be the quotient map. Any $(R/I)$-module $M$ becomes an $R$-module by restriction of scalars. In this $R$-module, every element of $I$ acts by $0$.
   (This links the notions of quotient ring and modules over it.)

3. Localization map.
   For a multiplicative set $S\subset R$, the localization map $R\to S^{-1}R$ lets any $S^{-1}R$-module be viewed as an $R$-module. In the restricted $R$-module, multiplication by each $s\in S$ becomes an invertible endomorphism (because $s$ acts invertibly in $S^{-1}R$).

Related knowls

• Ring maps: ring homomorphism
• Changing scalars: extension of scalars , tensor product
• Module basics: module