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.
Given any set S and ring R, there is a free R-module with basis S, which is called the free module on S or module of formal R-linear combinations of the elements of S. A free abelian group is precisely a free module over the ring
has invariant basis number, then by definition any two bases have the same cardinality.
For example, nonzero commutative rings have invariant basis number.
The cardinality of any (and therefore every) basis is called the rank of the free module
If this cardinality is finite, the free module is said to be free of finite rank, or free of rank n if the rank is known to be n. Let R be a ring.
Given a set E and ring R, there is a free R-module that has E as a basis: namely, the direct sum of copies of R indexed by E Explicitly, it is the submodule of the Cartesian product
(R is viewed as say a left module) that consists of the elements that have only finitely many nonzero components.
One can embed E into R(E) as a subset by identifying an element e with that of R(E) whose e-th component is 1 (the unity of R) and all the other components are zero.
Then each element of R(E) can be written uniquely as where only finitely many
It is called a formal linear combination of elements of E. A similar argument shows that every free left (resp.
right) R-module is isomorphic to a direct sum of copies of R as left (resp.
The free module R(E) may also be constructed in the following equivalent way.
Given a ring R and a set E, first as a set we let We equip it with a structure of a left module such that the addition is defined by: for x in E, and the scalar multiplication by: for r in R and x in E, Now, as an R-valued function on E, each f in
is a free module with the basis E. The inclusion mapping
defined above is universal in the following sense.
from a set E to a left R-module N, there exists a unique module homomorphism
The uniqueness means that each R-linear map
is uniquely determined by its restriction to E. As usual for universal properties, this defines R(E) up to a canonical isomorphism.
for each set E determines a functor from the category of sets to the category of left R-modules.
It is called the free functor and satisfies a natural relation: for each set E and a left module N, where
is a left adjoint of the forgetful functor.
Many statements true for free modules extend to certain larger classes of modules.
Flat modules are defined by the property that tensoring with them preserves exact sequences.
Torsion-free modules form an even broader class.
For a finitely generated module over a PID (such as Z), the properties free, projective, flat, and torsion-free are equivalent.
This article incorporates material from free vector space over a set on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.