Editing Free module (section)

== Universal property ==

The inclusion mapping <math>\iota : E\to R^{(E)}</math> defined above is [[universal property|universal]] in the following sense. Given an arbitrary function <math>f : E\to N</math> from a set {{math|''E''}} to a left {{math|''R''}}-module {{math|''N''}}, there exists a unique [[module homomorphism]] <math>\overline{f}: R^{(E)}\to N</math> such that <math>f = \overline{f} \circ\iota</math>; namely, <math>\overline{f}</math> is defined by the formula:
:<math>\overline{f}\left (\sum_{e \in E} r_e e \right) = \sum_{e \in E} r_e f(e)</math>
and <math>\overline{f}</math> is said to be obtained by ''extending <math>f</math> by linearity.'' The uniqueness means that each ''R''-linear map <math>R^{(E)} \to N</math> is uniquely determined by its [[Restriction (mathematics)|restriction]] to ''E''.

As usual for universal properties, this defines {{math|''R''<sup>(''E'')</sup>}} [[up to]] a [[canonical isomorphism]]. Also the formation of <math>\iota : E\to R^{(E)}</math> for each set ''E'' determines a [[functor]]
: <math>R^{(-)}: \textbf{Set} \to R\text{-}\mathsf{Mod}, \, E \mapsto R^{(E)}</math>,
from the [[category of sets]] to the category of left {{math|''R''}}-modules. It is called the [[free functor]] and satisfies a natural relation: for each set ''E'' and a left module ''N'',
: <math>\operatorname{Hom}_{\textbf{Set}}(E, U(N)) \simeq \operatorname{Hom}_R(R^{(E)}, N), \, f \mapsto \overline{f}</math>
where <math>U: R\text{-}\mathsf{Mod} \to \textbf{Set}</math> is the [[forgetful functor]], meaning <math>R^{(-)}</math> is a [[left adjoint]] of the forgetful functor.