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 rankn if the rank is known to be n.
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, If R is commutative, a nonzero ideal I of R is free if and only if it is a principal ideal generated by a nonzerodivisor, with a generator being a basis.
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).
A finitely generated module over a commutative local ring is free if and only if it is faithfully flat. Also, Kaplansky's theorem states a projective module over a (possibly non-commutative) local ring is free.
Sometimes, whether a module is free or not is undecidable in the set-theoretic sense. A famous example is the Whitehead problem, which asks whether a Whitehead group is free or not. As it turns out, the problem is independent of ZFC.
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 embedE 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
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.
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.
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.
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.
^Keown (1975). An Introduction to Group Representation Theory. p. 24.
^Hazewinkel (1989). Encyclopaedia of Mathematics, Volume 4. p. 110.
^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.