Definition

Suppose that ${\displaystyle M}$ is a module over ${\displaystyle S}$. Then it can be regarded as a module over ${\displaystyle R}$ where the action of ${\displaystyle R}$ is given via

${\displaystyle {\begin{aligned}M\times R&\longrightarrow M\\(m,r)&\longmapsto m\cdot f(r)\end{aligned}}}$

where ${\displaystyle m\cdot f(r)}$ denotes the action defined by the ${\displaystyle S}$-module structure on ${\displaystyle M}$.^[1]

Interpretation as a functor

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

${\displaystyle u(m\cdot r)=u(m\cdot f(r))=u(m)\cdot f(r)=u(m)\cdot r\,}$.

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

If ${\displaystyle R}$ is the ring of integers, then this is just the forgetful functor from modules to abelian groups.

Definition

Let ${\displaystyle f:R\to S}$ be a homomorphism between two rings, and let ${\displaystyle M}$ be a module over ${\displaystyle R}$. Consider the tensor product ${\displaystyle M^{S}=M\otimes _{R}S}$, where ${\displaystyle S}$ is regarded as a left ${\displaystyle R}$-module via ${\displaystyle f}$. Since ${\displaystyle S}$ is also a right module over itself, and the two actions commute, that is ${\displaystyle r\cdot (s\cdot s')=(r\cdot s)\cdot s'}$ for ${\displaystyle r\in R}$, ${\displaystyle s,s'\in S}$ (in a more formal language, ${\displaystyle S}$ is a ${\displaystyle (R,S)}$-bimodule), ${\displaystyle M^{S}}$ inherits a right action of ${\displaystyle S}$. It is given by ${\displaystyle (m\otimes s)\cdot s'=m\otimes ss'}$ for ${\displaystyle m\in M}$, ${\displaystyle s,s'\in S}$. This module is said to be obtained from ${\displaystyle M}$ through extension of scalars.

Informally, extension of scalars is "the tensor product of a ring and a module"; more formally, it is a special case of a tensor product of a bimodule and a module – the tensor product of an R-module with an ${\displaystyle (R,S)}$-bimodule is an S-module.

Examples

One of the simplest examples is complexification, which is extension of scalars from the real numbers to the complex numbers. More generally, given any field extension K < L, one can extend scalars from K to L. In the language of fields, a module over a field is called a vector space, and thus extension of scalars converts a vector space over K to a vector space over L. This can also be done for division algebras, as is done in quaternionification (extension from the reals to the quaternions).

More generally, given a homomorphism from a field or commutative ring R to a ring S, the ring S can be thought of as an associative algebra over R, and thus when one extends scalars on an R-module, the resulting module can be thought of alternatively as an S-module, or as an R-module with an algebra representation of S (as an R-algebra). For example, the result of complexifying a real vector space (R = R, S = C) can be interpreted either as a complex vector space (S-module) or as a real vector space with a linear complex structure (algebra representation of S as an R-module).

Interpretation as a functor

Extension of scalars can be interpreted as a functor from ${\displaystyle R}$-modules to ${\displaystyle S}$-modules. It sends ${\displaystyle M}$ to ${\displaystyle M^{S}}$, as above, and an ${\displaystyle R}$-homomorphism ${\displaystyle u:M\to N}$ to the ${\displaystyle S}$-homomorphism ${\displaystyle u^{S}:M^{S}\to N^{S}}$ defined by ${\displaystyle u^{S}=u\otimes _{R}{\text{id}}_{S}}$.