Change of rings

In algebra, given a ring homomorphism formula_1, there are three ways to change the coefficient ring of a module; namely, for a left "R"-module "M" and a left "S"-module "N",
They are related as adjoint functors:
and
This is related to Shapiro's lemma.
Operations.
Restriction of scalars.
Throughout this section, let formula_7 and formula_8 be two rings (they may or may not be commutative, or contain an identity), and let formula_9 be a homomorphism. Restriction of scalars changes "S"-modules into "R"-modules. In algebraic geometry, the term "restriction of scalars" is often used as a synonym for Weil restriction.
Definition.
Suppose that formula_10 is a module over formula_8. Then it can be regarded as a module over formula_7 where the action of formula_7 is given via
where formula_15 denotes the action defined by the formula_8-module structure on formula_10.
Interpretation as a functor.
Restriction of scalars can be viewed as a functor from formula_8-modules to formula_7-modules. An formula_8-homomorphism formula_21 automatically becomes an formula_7-homomorphism between the restrictions of formula_10 and formula_24. Indeed, if formula_25 and formula_26, then
As a functor, restriction of scalars is the right adjoint of the extension of scalars functor.
If formula_7 is the ring of integers, then this is just the forgetful functor from modules to abelian groups.
Extension of scalars.
Extension of scalars changes "R"-modules into "S"-modules.
Definition.
Let formula_29 be a homomorphism between two rings, and let formula_10 be a module over formula_7. Consider the tensor product formula_32, where formula_8 is regarded as a left formula_7-module via formula_35. Since formula_8 is also a right module over itself, and the two actions commute, that is formula_37 for formula_26, formula_39 (in a more formal language, formula_8 is a formula_41-bimodule), formula_42 inherits a right action of formula_8. It is given by formula_44 for formula_25, formula_39. This module is said to be obtained from formula_10 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 formula_41 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).
Applications.
This generalization is useful even for the study of fields – notably, many algebraic objects associated to a field are not themselves fields, but are instead rings, such as algebras over a field, as in representation theory. Just as one can extend scalars on vector spaces, one can also extend scalars on group algebras and also on modules over group algebras, i.e., group representations. Particularly useful is relating how irreducible representations change under extension of scalars – for example, the representation of the cyclic group of order 4, given by rotation of the plane by 90°, is an irreducible 2-dimensional "real" representation, but on extension of scalars to the complex numbers, it split into 2 complex representations of dimension 1. This corresponds to the fact that the characteristic polynomial of this operator, formula_49 is irreducible of degree 2 over the reals, but factors into 2 factors of degree 1 over the complex numbers – it has no real eigenvalues, but 2 complex eigenvalues.
Interpretation as a functor.
Extension of scalars can be interpreted as a functor from formula_7-modules to formula_8-modules. It sends formula_10 to formula_42, as above, and an formula_7-homomorphism formula_21 to the formula_8-homomorphism formula_57 defined by formula_58.
Relation between the extension of scalars and the restriction of scalars.
Consider an formula_7-module formula_10 and an formula_8-module formula_24. Given a homomorphism formula_63, define formula_64 to be the composition
where the last map is formula_66. This formula_67 is an formula_8-homomorphism, and hence formula_69 is well-defined, and is a homomorphism (of abelian groups).
In case both formula_7 and formula_8 have an identity, there is an inverse homomorphism formula_72, which is defined as follows. Let formula_73. Then formula_74 is the composition
where the first map is the canonical isomorphism formula_76.
This construction shows that the groups formula_77 and formula_78 are isomorphic. Actually, this isomorphism depends only on the homomorphism formula_35, and so is functorial. In the language of category theory, the extension of scalars functor is left adjoint to the restriction of scalars functor.