Editing Free module

{{short description|In mathematics, a module that has a basis}}
In [[mathematics]], a '''free module''' is a [[module (mathematics)|module]] that has a ''basis'', that is, a [[generating set of a module|generating set]] that is [[linearly independent]]. Every [[vector space]] is a free module,<ref>{{cite book|author=Keown |title=An Introduction to Group Representation Theory|year=1975|url={{Google books|plainurl=y|id=hC9iTw8DO7gC|page=24|text=Every vector space is free}}|page=24}}</ref> but, if the [[ring (mathematics)|ring]] of the coefficients is not a [[division ring]] (not a [[field (mathematics)|field]] in the [[commutative ring|commutative]] case), then there exist non-free modules.

Given any [[Set (mathematics)|set]] {{math|''S''}} and ring {{math|''R''}}, there is a free {{math|''R''}}-module with basis {{math|''S''}}, which is called the ''free module on'' {{math|''S''}} or ''module of formal'' {{math|''R''}}-''linear combinations'' of the elements of {{math|''S''}}.

A [[free abelian group]] is precisely a free module over the ring <math>\Z</math> of [[integer]]s.

== Definition ==

For a [[ring (mathematics)|ring]] <math>R</math> and an <math>R</math>-[[module (mathematics)|module]] <math>M</math>, the set <math>E\subseteq M</math> is a basis for <math>M</math> if:
* <math>E</math> is a [[generating set of a module|generating set]] for <math>M</math>; that is to say, every element of <math>M</math> is a finite sum of elements of <math>E</math> multiplied by coefficients in <math>R</math>; and
* <math>E</math> is [[linearly independent]]: for every set <math>\{e_1,\dots,e_n\}\subset E</math> of distinct elements, <math>r_1 e_1 + r_2 e_2 + \cdots + r_n e_n = 0_M</math> implies that <math>r_1 = r_2 = \cdots = r_n = 0_R</math> (where <math>0_M</math> is the zero element of <math>M</math> and <math>0_R</math> is the zero element of <math>R</math>).

A free module is a module with a basis.<ref>{{cite book|author=Hazewinkel |title=Encyclopaedia of Mathematics, Volume 4|year=1989|url={{Google books|plainurl=y|id=s9F71NJxwzoC|page=110|text=A free module is a module with a basis}}|page=110}}</ref>

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 <math>R</math> 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 <math>M</math>. If this cardinality is finite, the free module is said to be ''free of finite rank'', or ''free of rank'' {{mvar|n}} if the rank is known to be {{mvar|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.<ref>Proof: Suppose <math>I</math> is free with a basis <math>\{ x_j | j\}</math>. For <math>j \ne k</math>, <math>x_j x_k</math> must have the unique linear combination in terms of <math>x_j</math> and <math>x_k</math>, which is not true. Thus, since <math>I \ne 0</math>, there is only one basis element which must be a nonzerodivisor. The converse is clear.<math>\square</math></ref><!-- How about the non-commutative case? we at least need a reference for the non-commutative case. -->
* Over a [[principal ideal domain]] (e.g., <math>\mathbb{Z}</math>), a submodule of a free module is free.
* If ''R'' is commutative, the polynomial ring <math>R[X]</math> in indeterminate ''X'' is a free module with a possible basis 1, ''X'', ''X''<sup>2</sup>, ....
* Let <math>A[t]</math> be a polynomial ring over a commutative ring ''A'', ''f'' a monic polynomial of degree ''d'' there, <math>B = A[t]/(f)</math> and <math>\xi</math> the image of ''t'' in ''B''. Then ''B'' contains ''A'' as a subring and is free as an ''A''-module with a basis <math>1, \xi, \dots, \xi^{d-1}</math>.
* For any non-negative integer ''n'', <math>R^n = R \times \cdots \times R</math>, the [[Direct_product#Direct_product_of_modules|cartesian product]] of ''n'' copies of ''R'' as a left ''R''-module, is free. If ''R'' has [[invariant basis number]], then its [[rank of a module|rank]] is ''n''.
* A [[Direct sum of modules|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 [[Flat module#Faithful flatness|faithfully flat]].<ref>{{harvnb|Matsumura|1986|loc=Theorem 7.10.}}</ref> Also, [[Kaplansky's theorem on projective modules|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_problem#Examples_of_undecidable_statements|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 ==

{{anchor|Free module over a set}}Given a set {{math|''E''}} and ring {{math|''R''}}, there is a free {{math|''R''}}-module that has {{math|''E''}} as a basis: namely, the [[direct sum of modules|direct sum]] of copies of ''R'' indexed by ''E''
: <math>R^{(E)} = \bigoplus_{e \in E} R</math>.
Explicitly, it is the submodule of the [[Direct_product#Direct_product_of_modules|Cartesian product]] <math display="inline">\prod_E R</math> (''R'' is viewed as say a left module) that consists of the elements that have only finitely many nonzero components. One can [[Embedding|embed]] ''E'' into {{math|''R''<sup>(''E'')</sup>}} as a subset by identifying an element ''e'' with that of {{math|''R''<sup>(''E'')</sup>}} whose ''e''-th component is 1 (the unity of ''R'') and all the other components are zero. Then each element of {{math|''R''<sup>(''E'')</sup>}} can be written uniquely as
: <math>\sum_{e \in E} c_e e ,</math>
where only finitely many <math>c_e</math> are nonzero. It is called a ''[[formal linear combination]]'' of elements of {{math|''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 {{math|''R''<sup>(''E'')</sup>}} may also be constructed in the following equivalent way.

Given a ring ''R'' and a set ''E'', first as a set we let
: <math>R^{(E)} = \{ f: E \to R \mid f(x) = 0 \text { for all but finitely many } x \in E \}.</math>
We equip it with a structure of a left module such that the addition is defined by: for ''x'' in ''E'',
: <math>(f+g)(x) = f(x) + g(x)</math>
and the scalar multiplication by: for ''r'' in ''R'' and ''x'' in ''E'',
: <math>(r f)(x) = r f(x)</math>

Now, as an ''R''-valued [[Function (mathematics)|function]] on ''E'', each ''f'' in <math>R^{(E)}</math> can be written uniquely as
: <math>f = \sum_{e \in E} c_e \delta_e</math>
where <math>c_e</math> are in ''R'' and only finitely many of them are nonzero and <math>\delta_e</math> is given as
: <math> \delta_e(x) = \begin{cases} 1_R \quad\mbox{if } x=e \\ 0_R \quad\mbox{if } x\neq e \end{cases} </math>
(this is a variant of the [[Kronecker delta]]). The above means that the subset <math>\{ \delta_e \mid e \in E \}</math> of <math>R^{(E)}</math> is a basis of <math>R^{(E)}</math>. The mapping <math>e \mapsto \delta_e</math> is a [[bijection]] between {{math|''E''}} and this basis. Through this bijection, <math>R^{(E)}</math> is a free module with the basis ''E''.

== Universal property ==

The inclusion mapping <math>\iota : E\to R^{(E)}</math> defined above is [[universal property|universal]] in the following sense. Given an arbitrary function <math>f : E\to N</math> from a set {{math|''E''}} to a left {{math|''R''}}-module {{math|''N''}}, there exists a unique [[module homomorphism]] <math>\overline{f}: R^{(E)}\to N</math> such that <math>f = \overline{f} \circ\iota</math>; namely, <math>\overline{f}</math> is defined by the formula:
:<math>\overline{f}\left (\sum_{e \in E} r_e e \right) = \sum_{e \in E} r_e f(e)</math>
and <math>\overline{f}</math> is said to be obtained by ''extending <math>f</math> by linearity.'' The uniqueness means that each ''R''-linear map <math>R^{(E)} \to N</math> is uniquely determined by its [[Restriction (mathematics)|restriction]] to ''E''.

As usual for universal properties, this defines {{math|''R''<sup>(''E'')</sup>}} [[up to]] a [[canonical isomorphism]]. Also the formation of <math>\iota : E\to R^{(E)}</math> for each set ''E'' determines a [[functor]]
: <math>R^{(-)}: \textbf{Set} \to R\text{-}\mathsf{Mod}, \, E \mapsto R^{(E)}</math>,
from the [[category of sets]] to the category of left {{math|''R''}}-modules. It is called the [[free functor]] and satisfies a natural relation: for each set ''E'' and a left module ''N'',
: <math>\operatorname{Hom}_{\textbf{Set}}(E, U(N)) \simeq \operatorname{Hom}_R(R^{(E)}, N), \, f \mapsto \overline{f}</math>
where <math>U: R\text{-}\mathsf{Mod} \to \textbf{Set}</math> is the [[forgetful functor]], meaning <math>R^{(-)}</math> is a [[left adjoint]] of the forgetful functor.

== Generalizations ==

Many statements true for free modules extend to certain larger classes of modules. [[Projective module]]s are direct summands of free modules.  [[Flat module]]s are defined by the property that tensoring with them preserves exact sequences.  [[Torsion-free module]]s 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.

: [[File:Module properties in commutative algebra.svg|Module properties in commutative algebra]]

See [[local ring]], [[perfect ring]] and [[Dedekind ring]].

== See also ==
* [[Free object]]
* [[free presentation]]
* [[free resolution]]
* [[Quillen–Suslin theorem]]
* [[stably free module]]
* [[generic freeness]]

== Notes ==
{{reflist}}

== References ==
{{PlanetMath attribution|id=4196|title=free vector space over a set}}
* {{cite book | first= Iain T.|last= Adamson | title=Elementary Rings and Modules | series=University Mathematical Texts | publisher=Oliver and Boyd | year=1972 | isbn=0-05-002192-3 | pages=65–66|mr=0345993}}
* {{cite book |last1=Keown |first1=R. |title=An Introduction to Group Representation Theory |series=Mathematics in science and engineering |volume=116 |year=1975 |publisher=Academic Press |isbn=978-0-12-404250-6 |mr=0387387 }}
* {{SpringerEOM | title=Free module | id=Free_module&oldid=13029  | last=Govorov | first=V. E. }}.
* {{cite book
 |last1      = Matsumura
 |first1     = Hideyuki
 |author-link1= Hideyuki Matsumura
 |year       = 1986
 |title      = Commutative ring theory
 |series     = Cambridge Studies in Advanced Mathematics
 |volume     = 8
 |url        = {{google books|yJwNrABugDEC|Commutative ring theory|plainurl=yes|page=123}}
 |publisher  = Cambridge University Press
 |isbn       = 0-521-36764-6
 |mr         = 0879273
 |zbl         = 0603.13001
}}

{{Dimension topics}}

[[Category:Module theory]]
[[Category:Free algebraic structures]]