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 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.
- 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 every subset of 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.
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.
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.
- 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.)
- Kaplansky's theorem states a projective module over a local ring is free.
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.
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 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.
- 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.
- Adamson, Iain T. (1972). Elementary Rings and Modules. University Mathematical Texts. Oliver and Boyd. pp. 65–66. ISBN 0-05-002192-3. MR 0345993.
- Keown, R. (1975). An Introduction to Group Representation Theory. Mathematics in science and engineering. 116. Academic Press. ISBN 978-0-12-404250-6. MR 0387387.
- Govorov, V. E. (2001) , "Free module", Encyclopedia of Mathematics, EMS Press.