Editing Basis (linear algebra) (section)

===Free module===
{{main|Free module|Free abelian group}} 
If one replaces the field occurring in the definition of a vector space by a [[ring (mathematics)|ring]], one gets the definition of a [[module (mathematics)|module]]. For modules, [[linear independence]] and [[spanning set]]s are defined exactly as for vector spaces, although "[[generating set of a module|generating set]]" is more commonly used than that of "spanning set".

Like for vector spaces, a ''basis'' of a module is a linearly independent subset that is also a generating set. A major difference with the theory of vector spaces is that not every module has a basis. A module that has a basis is called a ''free module''. Free modules play a fundamental role in module theory, as they may be used for describing the structure of non-free modules through [[free resolution]]s.

A module over the integers is exactly the same thing as an [[abelian group]]. Thus a free module over the integers is also a free abelian group. Free abelian groups have specific properties that are not shared by modules over other rings. Specifically, every subgroup of a free abelian group is a free abelian group, and, if {{mvar|G}} is a subgroup of a finitely generated free abelian group {{mvar|H}} (that is an abelian group that has a finite basis), then there is a basis <math>\mathbf e_1, \ldots, \mathbf e_n</math> of {{mvar|H}} and an integer {{math|0 ≤ ''k'' ≤ ''n''}} such that <math>a_1 \mathbf e_1, \ldots, a_k \mathbf e_k</math> is a basis of {{mvar|G}}, for some nonzero integers {{nowrap|<math>a_1, \ldots, a_k</math>.}} For details, see {{slink|Free abelian group|Subgroups}}.