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In mathematics, a free module is a module that has a basis, that is, a generating set that is linearly independent. Every vector space is a free module, [ 1 ] 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.
However this module has a torsion-free rank equal to 1. If R is any ring and n a natural number, then the cartesian product R n is both a left and right R-module over R if we use the component-wise operations. Hence when n = 1, R is an R-module, where the scalar multiplication is just ring
A free module is a module that can be represented as a direct sum over its base ring, so free abelian groups and free -modules are equivalent concepts: each free abelian group is (with the multiplication operation above) a free -module, and each free -module comes from a free abelian group in this way. [21]
Then one finds a free module and a surjective homomorphism : (). Continuing in this fashion, a sequence of free modules F n {\displaystyle F_{n}} and homomorphisms p n {\displaystyle p_{n}} can be defined.
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 resolutions. 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.
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For example, modules need not have bases, as the Z-module (that is, abelian group) Z/2Z shows; those modules that do (including all vector spaces) are known as free modules. Nevertheless, a vector space can be compactly defined as a module over a ring which is a field , with the elements being called vectors.
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