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 𝑅 generated by a set 𝑆. The first is

𝑅(𝑆)=span⁡{1𝑠:𝑠∈𝑆⟩}≤𝑅𝑆

where we think of elements as maps of finite support 𝑆 →𝑅, and we identify 𝑠 ∈𝑆 with 1𝑠 :𝑡 ↦[𝑠 =𝑡]. The second is

𝑅⋅{𝑥𝑠}𝑠∈𝑆=span𝑅⁡{𝑥𝑠:𝑠∈𝑆}

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

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

Universal property

Let 𝑅 be a ring and 𝑆 be a set. The free module is a pair consisting of an 𝑅-module 𝑅(𝑆) and a function 𝜄 :𝑆 →𝑅(𝑆) such that given any 𝑅-module 𝑀 and function 𝑓 :𝑆 →𝑀 there exists a unique module homomorphism ¯𝑓 :𝑅(𝑆) →𝑀 such that the following diagram commutes

This has a unique extension to a functor such that

\begin{align*} \iota : 1 \Rightarrow R^{(-)}: \Set \to \Set \end{align*}

becomes a natural transformation.

Monoidal functor

When 𝑅 =K 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 𝟙 ={ ∙} in \Set, then we let

𝛾:K→K{∙}1↦1∙𝜇𝑋,𝑌:K(𝑋)⊗K(𝑌)→K(𝑋×𝑌)1𝑥⊗1𝑦↦1(𝑥,𝑦)

Construction as maps

Let 𝑆 be a set and 𝑅 be a ring. The free module 𝑅(𝑆) is the set of maps of finite support 𝑆 →𝑅 with addition and scaling induced by those of 𝑅, module
i.e. for all 𝑠 ∈𝑆

(𝛼𝑎+𝛽𝑏)(𝑠)=𝛼𝑎(𝑠)+𝛽𝑏(𝑠)

where we identify 𝑠 ∈𝑆 with 1𝑠 :𝑡 ↦[𝑡 =𝑠] invoking an Iverson bracket.

Proof of universal property

Clearly 𝑅(𝑆) as constructed is an 𝑅-module with basis {1𝑠}𝑠∈𝑆 Now let 𝑀 be an 𝑅-module and 𝑓 :𝑆 →𝑀 be a function. For a module homomorphism ¯𝑓 :𝑅(𝑆) →𝑀 to make the diagram commute, we require that ¯𝑓(1𝑠) =𝑓(𝑠) for all 𝑠 ∈𝑆, which fully specifies ¯𝑓 so that for 𝑎 ∈𝑅(𝑆)

¯𝑓(𝑎)=∑𝑠∈𝑆𝑓(𝑠)𝑎(𝑠)

as required.

Properties

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

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