Free vector space

In mathematics, a free module is a module that has a basis – that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in the commutative case), then there exist non-free modules.

Given any set S and ring R, there is a free R-module with basis S, which is called free module on S or module of formal linear combinations of the elements of S.

A free abelian group is precisely a free module over the ring Z of integers.

For a ring ${\displaystyle R}$ and an ${\displaystyle R}$-module ${\displaystyle M}$, the set ${\displaystyle E\subseteq M}$ is a basis for ${\displaystyle M}$ if:

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