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 nonfree modules.
Given any set S and ring R, there is a free Rmodule with basis S, which is called the free module on S or module of formal Rlinear combinations of the elements of S.
A free abelian group is precisely a free module over the ring Z of integers.
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Module theory  Lecture 6  Vector spaces vs Free modules

Torsion module, Torsion free modules Every vector space is torsion free. M/T is torsion Free.

FREE MODULES AND VECTOR SPACES OVER A DIVISION RING

What is a Module? (Abstract Algebra)

Rings 8 Free modules
Transcription
Definition
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 if for every of distinct elements, 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. 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.
Examples
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.^{[3]}
 Over a principal ideal domain (e.g., ), a submodule of a free module is free.
 If R is commutative, the polynomial ring in indeterminate X is a free module with a possible basis 1, X, X^{2}, ....
 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 Amodule with a basis .
 For any nonnegative integer n, , the cartesian product of n copies of R as a left Rmodule, is free. If R has invariant basis number, 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).
 A finitely generated module over a commutative local ring is free if and only if it is faithfully flat.^{[4]} Also, Kaplansky's theorem states a projective module over a (possibly noncommutative) local ring is free.
 Sometimes, whether a module is free or not is undecidable in the settheoretic 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 combinations
Given a set E and ring R, there is a free Rmodule 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 eth 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) Rmodule 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 Rvalued 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 Rmodule 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 Rlinear 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 Rmodules. 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.
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 torsionfree modules. If the ring has special properties, this hierarchy may collapse, e.g., for any perfect local Dedekind ring, every torsionfree module is flat, projective and free as well. A finitely generated torsionfree module of a commutative PID is free. A finitely generated Zmodule is free if and only if it is flat.
See local ring, perfect ring and Dedekind ring.
See also
 Free object
 Projective object
 free presentation
 free resolution
 Quillen–Suslin theorem
 stably free module
 generic freeness
Notes
 ^ 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.
 ^ Matsumura 1986, Theorem 7.10.
References
This article incorporates material from free vector space over a set on PlanetMath, which is licensed under the Creative Commons Attribution/ShareAlike License.
 Adamson, Iain T. (1972). Elementary Rings and Modules. University Mathematical Texts. Oliver and Boyd. pp. 65–66. ISBN 0050021923. MR 0345993.
 Keown, R. (1975). An Introduction to Group Representation Theory. Mathematics in science and engineering. Vol. 116. Academic Press. ISBN 9780124042506. MR 0387387.
 Govorov, V. E. (2001) [1994], "Free module", Encyclopedia of Mathematics, EMS Press.
 Matsumura, Hideyuki (1986). Commutative ring theory. Cambridge Studies in Advanced Mathematics. Vol. 8. Cambridge University Press. ISBN 0521367646. MR 0879273. Zbl 0603.13001.