Editing Free module (section)

{{short description|In mathematics, a module that has a basis}}
In [[mathematics]], a '''free module''' is a [[module (mathematics)|module]] that has a ''basis'', that is, a [[generating set of a module|generating set]] that is [[linearly independent]]. Every [[vector space]] is a free module,<ref>{{cite book|author=Keown |title=An Introduction to Group Representation Theory|year=1975|url={{Google books|plainurl=y|id=hC9iTw8DO7gC|page=24|text=Every vector space is free}}|page=24}}</ref> but, if the [[ring (mathematics)|ring]] of the coefficients is not a [[division ring]] (not a [[field (mathematics)|field]] in the [[commutative ring|commutative]] case), then there exist non-free modules.

Given any [[Set (mathematics)|set]] {{math|''S''}} and ring {{math|''R''}}, there is a free {{math|''R''}}-module with basis {{math|''S''}}, which is called the ''free module on'' {{math|''S''}} or ''module of formal'' {{math|''R''}}-''linear combinations'' of the elements of {{math|''S''}}.

A [[free abelian group]] is precisely a free module over the ring <math>\Z</math> of [[integer]]s.