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 (not a field in the commutative case), 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 formula_1 and an formula_1-module formula_3, the set formula_4 is a basis for formula_3 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 formula_1 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 formula_3. If this cardinality is finite, the free module is said to be "free of finite rank", or "free of rank" if the rank is known to be .
Examples.
Let "R" be a ring.
Formal linear combinations.
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 formula_29 ("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 as a subset by identifying an element "e" with that of whose "e"-th component is 1 (the unity of "R") and all the other components are zero. Then each element of can be written uniquely as
where only finitely many formula_31 are nonzero. It is called a "formal linear combination" of elements of .
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. right) 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 formula_35 can be written uniquely as
where formula_31 are in "R" and only finitely many of them are nonzero and formula_38 is given as
(this is a variant of the Kronecker delta.) The above means that the subset formula_40 of formula_35 is a basis of formula_35. The mapping formula_43 is a bijection between and this basis. Through this bijection, formula_35 is a free module with the basis "E".
Universal property.
The inclusion mapping formula_45 defined above is universal in the following sense. Given an arbitrary function formula_46 from a set to a left -module , there exists a unique module homomorphism formula_47 such that formula_48; namely, formula_49 is defined by the formula:
and formula_49 is said to be obtained by "extending formula_52 by linearity." The uniqueness means that each "R"-linear map formula_53 is uniquely determined by its restriction to "E".
As usual for universal properties, this defines up to a canonical isomorphism. Also the formation of formula_45 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 formula_57 is the forgetful functor, meaning formula_58 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.