Free module

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 field, there exist non-free modules.

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

A free module is a module with a basis: a linearly independent generating set.

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

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