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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.
In fact each faithful K[G]-module is a faithful representation of G, but the converse does not hold. Consider for example the natural representation of the symmetric group S n in n dimensions by permutation matrices, which is certainly faithful. Here the order of the group is n! while the n × n matrices form a vector space of dimension n 2.
[2] On a local ring every finitely generated flat module is free. [3] A finitely generated flat module that is not projective can be built as follows. Let = be the set of the infinite sequences whose terms belong to a fixed field F. It is a commutative ring with addition and multiplication defined componentwise.
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]
SageMath is designed partially as a free alternative to the general-purpose mathematics products Maple and MATLAB. It can be downloaded or used through a web site. SageMath comprises a variety of other free packages, with a common interface and language. SageMath is developed in Python.
A right R-module M R is defined similarly in terms of an operation · : M × R → M. Authors who do not require rings to be unital omit condition 4 in the definition above; they would call the structures defined above "unital left R-modules". In this article, consistent with the glossary of ring theory, all rings and modules are assumed to be ...
The category K-Vect (some authors use Vect K) has all vector spaces over a field K as objects, and K-linear maps as morphisms. Since vector spaces over K (as a field) are the same thing as modules over the ring K, K-Vect is a special case of R-Mod (some authors use Mod R), the category of left R-modules.
2. A semifir is a ring in which every finitely generated right ideal is a free module of fixed rank. 3. The free product of a family of associative is an associative algebra obtained, roughly, by the generators and the relations of the algebras in the family. The notion depends on which category of associative algebra is considered; for example ...