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 division ring, then there exist non-free modules.
Given any set and ring, there is a free -module with basis, which is called the free module on or module of formal -linear combinations of the elements of.
A free abelian group is precisely a free module over the ring of integers.

Definition

For a ring and an -module, the set is a basis for if:
A free module is a module with a basis.
An immediate consequence of the second half of the definition is that the coefficients in the first half are unique for each element of M.
If has invariant basis number, then by definition any two bases have the same cardinality. The cardinality of any basis is called the rank of the free module. If this cardinality is finite, the free module is said to be free of rank n, or simply free of finite rank.

Examples

Let R be a ring.
Given a set and ring, there is a free -module that has as a basis: namely, the direct sum of copies of R indexed by E
Explicitly, it is the submodule of the cartesian product that consists of the elements that have only finitely many nonzero components. One can embed E into as a subset by identifying an element e with that of whose e-th component is 1 and all the other components are zero. Then each element of can be written uniquely as
where only finitely many are nonzero. It is called a formal linear combination of elements of.
A similar argument shows that every free left R-module is isomorphic to a direct sum of copies of R as left module.

Another construction

The free module 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 can be written uniquely as
where are in R and only finitely many of them are nonzero and is given as
The above means that the subset of is a basis of. The mapping is a bijection between and this basis. Through this bijection, is a free module with the basis E.

Universal property

The inclusion mapping defined above is universal in the following sense. Given an arbitrary function from a set to a left -module, there exists a unique module homomorphism such that ; namely, is defined by the formula:
and is said to be obtained by extending by linearity. The uniqueness means that each R-linear map is uniquely determined by its restriction to E.
As usual for universal properties, this defines up to a canonical isomorphism. Also the formation of for each set E determines a functor
from the category of sets to the category of left -modules. It is called the free functor and satisfies a natural relation: for each set E and a left module N,
where is the forgetful functor, meaning is a left adjoint of the forgetful functor.

Generalizations

Many statements about free modules, which are wrong for general modules over rings, are still true for certain generalisations of free modules. Projective modules are direct summands of free modules, so one can choose an injection into a free module and use the basis of this one to prove something for the projective module. Even weaker generalisations are flat modules, which still have the property that tensoring with them preserves exact sequences, and torsion-free modules. If the ring has special properties, this hierarchy may collapse, e.g., for any perfect local Dedekind ring, every torsion-free module is flat, projective and free as well. A finitely generated torsion-free module of a commutative PID is free. A finitely generated Z-module is free if and only if it is flat.
See local ring, perfect ring and Dedekind ring.