Connection With Restriction of Scalars
Consider an -module and an -module . Given a homomorphism, where is viewed as an -module via restriction of scalars, define to be the composition
- ,
where the last map is . This is an -homomorphism, and hence is well-defined, and is a homomorphism (of abelian groups).
In case both and have an identity, there is an inverse homomorphism, which is defined as follows. Let . Then is the composition
- ,
where the first map is the canonical isomorphism .
This construction shows that the groups and are isomorphic. Actually, this isomorphism depends only on the homomorphism, and so is functorial. In the language of category theory, the extension of scalars functor is left adjoint to the restriction of scalars functor.
Read more about this topic: Extension Of Scalars
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