Let be a commutative ring with unity. A free module over is a (unital) module isomorphic to a direct sum of copies of . In particular, as every abelian group is a -module, a free abelian group is a direct sum of copies of . This is equivalent to saying that the module has a free basis, i.e. a set of elements with the that every element of the module can be uniquely expressed as an linear combination over of elements of the free basis. In the case that a free module over is a sum of finitely many copies of , then the number of copies is called the rank of the free module.
An alternative definition of a free module is via its universal property: Given a set , the free -module on the set is equipped with a function satisfying the property that for any other -module and any function , there exists a unique -module map such that .
|Date of creation||2013-03-22 12:10:10|
|Last modified on||2013-03-22 12:10:10|
|Last modified by||mathcam (2727)|
|Defines||free abelian group|
|Defines||rank of a free module|