Free module explained
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 and ring, there is a free -module with basis, which is called the free module on or module of formal -linear combinations of the elements of .
A free abelian group is precisely a free module over the ring of integers.
Definition
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 if the rank is known to be .
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,
X2, ....
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, 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. 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 combinations
Given a set and ring, there is a free -module that has 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 as a subset by identifying an element
e with that of whose
e-th component is 1 (the unity of
R) and all the other components are zero. Then each element of can be written uniquely as
where only finitely many
are nonzero. It is called a
formal linear combination of elements of .
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 may also be constructed in the following equivalent way.
Given a ring R and a set E, first as a set we let
R(E)=\{f:E\toR\midf(x)=0forallbutfinitelymanyx\inE\}.
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
\deltae(x)=\begin{cases}1R ifx=e\ 0R ifx ≠ e\end{cases}
(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 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 to a left -module, there exists a unique
module homomorphism
such that
; namely,
is defined by the formula:
\overline{f}\left(\sumeree\right)=\sumeref(e)
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 up to a canonical isomorphism. Also the formation of
for each set
E determines a
functorR(-):bf{Set}\toR-Mod,E\mapstoR(E)
,from the
category of sets to the category of left -modules. It is called the free functor and satisfies a natural relation: for each set
E and a left module
N,
}(E, U(N)) \simeq \operatorname_R(R^, N), \, f \mapsto \overlinewhere
is the
forgetful functor, meaning
is a
left adjoint of the forgetful functor.
Generalizations
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.
See local ring, perfect ring and Dedekind ring.
See also
Notes
- Book: Keown . [{{Google books|plainurl=y|id=hC9iTw8DO7gC|page=24|text=Every vector space is free}} An Introduction to Group Representation Theory]. 1975. 24.
- Book: Hazewinkel . [{{Google books|plainurl=y|id=s9F71NJxwzoC|page=110|text=A free module is a module with a basis}} Encyclopaedia of Mathematics, Volume 4]. 1989. 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.
References
- Book: Adamson, Iain T.. Elementary Rings and Modules . University Mathematical Texts . Oliver and Boyd . 1972 . 0-05-002192-3 . 65–66. 0345993.
- Book: Keown . R. . An Introduction to Group Representation Theory . Mathematics in science and engineering . 116 . 1975 . Academic Press . 978-0-12-404250-6 . 0387387 .
- .
- Book: Matsumura. Hideyuki. 1986. [{{google books|yJwNrABugDEC|Commutative ring theory|plainurl=yes|page=123}}
Commutative ring theory
]. Cambridge Studies in Advanced Mathematics. 8. Cambridge University Press. 0-521-36764-6. 0879273. 0603.13001.