Restriction of scalars

In abstract algebra, restriction of scalars is a procedure of creating a module over a ring $R$ from a module over another ring $S$, given a homomorphism $f\; :\; R\; o\; S$ between them. Intuitively speaking, the resulting module "remembers" less information than the initial one, hence the name.

Definition

Let $R$ and $S$ be two rings (they may or may not be commutative, or contain an identity), and let $f:R\; o\; S$ be a homomorphism. Suppose that $M$ is a module over $S$. Then it can be regarded as a module over $R$, if the action of $R$ is given via $r\; cdot\; m\; =\; f(r)\; cdot\; m$ for $r\; in\; R$ and $m\; in\; M$.

Interpretation as a functor

Restriction of scalars can be viewed as a functor from $S$-modules to $R$-modules. An $S$-homomorphism $u\; :\; M\; o\; N$ automatically becomes an $R$-homomorphism between the restrictions of $M$ and $N$. Indeed, if $m\; in\; M$ and $r\; in\; R$, then

: $u(r\; cdot\; m)\; =\; u(f(r)\; cdot\; m)\; =\; f(r)\; cdot\; u(m)\; =\; rcdot\; u(m),$.

As a functor, restriction of scalars is the right adjoint of the extension of scalars functor.

The case of fields

When both $R$ and $S$ are fields, $f$ is necessarily a monomorphism, and so identifies $R$ with a subfield of $S$. In such a case an $S$-module is simply a vector space over $S$, and naturally over any subfield thereof. The module obtained by restriction is then simply a vector space over the subfield $R\; subset\; S$.