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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,[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 linear combinations of the elements of S.

A free abelian group is precisely a free module over the ring Z of integers.

Contents

DefinitionEdit

For a ring   and an  -module  , the set   is a basis for   if:

  •   is a generating set for  ; that is to say, every element of   is a finite sum of elements of   multiplied by coefficients in  ; and
  •   is linearly independent, that is,   for   distinct elements of   implies that   (where   is the zero element of   and   is the zero element of  ).

A free module is a module with a basis.[2]

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 (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 rank n, or simply free of finite rank.

ExamplesEdit

Let R be a ring.

  • R is a free module of rank one over itself (either as a left or right module); any unit element is a basis.
  • More generally, a (say) left ideal I of R is free if and only if it is a principal ideal generated by a left nonzerodivisor, with a generator being a basis.
  • If R is commutative, the polynomial ring   in indeterminate X is a free module with a possible basis 1, X, X2, ....
  • Let   be a polynomial ring over a commutative ring A, f a monic polynomial of degree d there,   and   the image of t in B. Then B contains A as a subring and is free as an A-module with a basis  .
  • For any non-negative integer n,  , the cartesian product of n copies of R as a left R-module, is free. If R has invariant basis number (which is true for commutative R), then its rank is n.
  • A direct sum of free modules is free, while an infinite cartesian product of free modules is generally not free (cf. the Baer–Specker group.)

Formal linear combinationsEdit

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   are nonzero. 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. right) module.

Another constructionEdit

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   can be written uniquely as

 

where   are in R and only finitely many of them are nonzero and   is given as

 

(this is a variant of the Kronecker delta.) The above means that the subset   of   is a basis of  . The mapping   is a bijection between E and this basis. Through this bijection,   is a free module with the basis E.

Universal propertyEdit

The inclusion mapping   defined above is universal in the following sense. Given an arbitrary function   from a set E to a left R-module N, 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 R(E) 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 R-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.

GeneralizationsEdit

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 in 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.

See alsoEdit

NotesEdit

  1. ^ Keown (1975). An Introduction to Group Representation Theory. p. 24.
  2. ^ Hazewinkel (1989). Encyclopaedia of Mathematics, Volume 4. p. 110.

ReferencesEdit

External linksEdit

This article incorporates material from free vector space over a set on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.