# Free module

In mathematics, a **free module** is a module having a *free basis*.

For an *R*-module *M*, the set *E* = {*e*_{1}, *e*_{2}, ... *e*_{n}} is a free basis for *M* if and only if:

*E*is a generating set for*M*, that is to say every element of*M*is a sum of elements of*E*multiplied by coefficients in*R*;- if
*r*_{1}*e*_{1}+*r*_{2}*e*_{2}+ ... +*r*_{n}*e*_{n}=**0**, then*r*_{1}=*r*_{2}= ... =*r*_{n}=*0*(where**0**is the zero element of*M*and*0*is the zero element of*R*).

If *M* has a free basis with *n* elements, then *M* is said to be *free of rank n*, or more generally *free of finite rank*.

Note that an immediate corollary of (2) is that the coefficients in (1) are unique for each *x*.

The definition of an infinite free basis is similar, except that *E* will have infinitely many elements. However the sum must be finite, and thus for any particular *x* only finitely many of the elements of *E* are involved.

In the case of an infinite basis, the rank of *M* is the cardinality of *E*.

## Construction

Given a set *E*, we can construct a free *R*-module over *E*, denoted by *C*(*E*), as follows:

**As a set,***C*(*E*) contains the functions*f*:*E*→*R*such that*f*(*x*) = 0 for all but finitely many*x*in*E*.**Addition:**for two elements*f*,*g*∈*C*(*E*), we define*f*+*g*∈*C*(*E*) by (*f*+*g*)(*x*) =*f*(*x*) +*g*(*x*) for all*x*∈*E*.**Scalar multiplication:**for α ∈*R*and*f*∈*C*(*E*), we define α*f*∈*C*(*E*) by (α*f*)(*x*) = α*f*(*x*) for all*x*∈*E*.

A basis for *C*(*E*) is given by the set { Δ_{a} : *a* ∈ *E* } where

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta_a(x) = \begin{cases} 1, \quad\mbox{if } x=a; \\ 0, \quad\mbox{if } x\neq a. \end{cases} }**

Define the mapping ι : *E* → *C*(*E*) by ι(*a*) = Δ_{a}. This mapping gives a bijection between *E* and the basis vectors {Δ_{a}}_{a∈X}. We can thus identify these spaces. Then *E* becomes a linearly independent basis for *C*(*E*).

## Universal property

The mapping ι : *E* → *C*(*E*) defined above is universal in the following sense. If φ is an arbitrary mapping from *E* to some *R*-module *M*, then there exists a unique mapping ψ *C*(*E*) → *M* such that φ = ψ o ι.

*This article incorporates material from free vector space over a set on PlanetMath, which is licensed under the GFDL.*
es:Módulo libre