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.