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 R-linear combinations of the elements of S.

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


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

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. 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.

Formal linear combinations

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 construction

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 property

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.


Many statements true for free modules extend to certain larger classes of modules. Projective modules are direct summands of free 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.

Module properties in commutative algebra

See local ring, perfect ring and Dedekind ring.

See also


  1. ^ Keown (1975). An Introduction to Group Representation Theory. p. 24.
  2. ^ Hazewinkel (1989). Encyclopaedia of Mathematics, Volume 4. p. 110.
  3. ^ Proof: Suppose is free with a basis . For , must have the unique linear combination in terms of and , which is not true. Thus, since , there is only one basis element which must be a nonzerodivisor. The converse is clear.
  4. ^ Matsumura 1986, Theorem 7.10.


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