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In abstract algebra, extension of scalars is a means of producing a module over a ring from a module over another ring , given a homomorphism between them. Intuitively, the new module admits multiplication by more scalars than the original one, hence the name extension.

Definition

In this definition the rings are assumed to be associative, but not necessarily commutative, or to have an identity. Also, modules are assumed to be left modules. The modifications needed in the case of right modules are straightforward.

Let be a homomorphism between two rings, and let be a module over . Consider the tensor product , where is regarded as a right -module via . Since is also a left module over itself, and the two actions commute, that is for , (in a more formal language, is a -bimodule), inherits a left action of . It is given by for and . This module is said to be obtained from through extension of scalars.

Interpretation as a functor

Extension of scalars can be interpreted as a functor from -modules to -modules. It sends to , as above, and an -homomorphism to the -homomorphism defined by .

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.

References