Restriction of scalars
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In abstract algebra, restriction of scalars is a procedure of creating a module over a ring from a module over another ring , given a homomorphism between them. Intuitively speaking, the resulting module "remembers" less information than the initial one, hence the name.
Let and be two rings (they may or may not be commutative, or contain an identity), and let be a homomorphism. Suppose that is a module over . Then it can be regarded as a module over , if the action of is given via for and .
Interpretation as a functor
Restriction of scalars can be viewed as a functor from -modules to -modules. An -homomorphism automatically becomes an -homomorphism between the restrictions of and . Indeed, if and , then
If is the ring of integers, then this is just the forgetful functor from modules to abelian groups.
The case of fields
When both and are fields, is necessarily a monomorphism, and so identifies with a subfield of . In such a case an -module is simply a vector space over , and naturally over any subfield thereof. The module obtained by restriction is then simply a vector space over the subfield .
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