Module theory MOC

Free module

Free modules are the free objects in Category of left modules. module In essence it is a module with a basis, hence every vector space is a free module, but one can form non-free modules over a ring that is not a Division ring.

Notation

In these notes, we have two conventions for the free module over generated by a set . The first is

where we think of elements as maps of finite support , and we identify with . The second is

which allows for the explicit naming of the basis to be used.

By basis for an -module , we mean an -spanning set such that each is given by a unique -linear combination of elements.

Universal property

Let be a ring and be a set. The free module is a pair consisting of an -module and a function such that given any -module and function there exists a unique module homomorphism such that the following diagram commutes

This has a unique extension to a functor such that

becomes a natural transformation.

Monoidal functor

When is a commutative ring this forms a monoidal functor with respect to the cartesian structure on Category of sets and the tensor product on Category of modules over a commutative ring. cat If in , then we let

Construction as maps

Let be a set and be a ring. The free module is the set of maps of finite support with addition and scaling induced by those of , module
i.e. for all

where we identify with invoking an Iverson bracket.

Proof of universal property

Clearly as constructed is an -module with basis Now let be an -module and be a function. For a module homomorphism to make the diagram commute, we require that for all , which fully specifies so that for

as required.

Properties

• carries the additional structure of an -comonoid, namely the Free R-comonoid

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