Free module

In the mathematical subfield of algebra , a free module is a module that has a basis . The concept of the free module is thus a generalization of the concepts of vector space or free Abelian group .

definition

A family of elements of a module (or more generally of a link module) over a ring is called linearly independent or free if for every finite index set and all the following applies: ${\displaystyle B:=\{b_{i}\mid i\in I\}}$ ${\displaystyle F}$ ${\displaystyle R}$ ${\displaystyle \textstyle J\subseteq I}$ ${\displaystyle \textstyle r_{i}\in R}$

${\displaystyle \sum _{i\in J}r_{i}\cdot b_{i}=0\;\Rightarrow \;\forall i\in J\colon \,r_{i}=0.}$

If they generate the module at the same time , a base (of ) and the module is called the free module above or simply free . ${\displaystyle \{b_{i}\mid i\in I\}}$ ${\displaystyle F}$ ${\displaystyle B}$ ${\displaystyle F}$ ${\displaystyle F}$ ${\displaystyle R}$ ${\displaystyle B}$

Remarks

First examples and counterexamples

1. Each ring with one element is free about itself. That is, is free legal module. A free link module is accordingly . ${\displaystyle R}$ ${\displaystyle R_{R}}$ ${\displaystyle _{R}R}$
2. If the module is not free. The module is torsion-free , but not free (free modules are always torsion-free). ${\displaystyle 1<n\in \mathbb {N} }$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {Q} }$
3. Is a natural number, then is a free module. The family is a basis . The -th component is the same , all other components are . This example is subordinate to the following situation: If there is an arbitrary set and a family of modules, then the coproduct is free if and only if all are free. In particular, is free. ${\displaystyle n}$ ${\displaystyle R^{n}=\left\{{\begin{pmatrix}r_{1},\dots ,r_{n}\end{pmatrix}}\mid r_{1},\dots r_{n}\in R\right\}}$ ${\displaystyle (e_{i}\mid i\in \{1,\dots ,n\})}$ ${\displaystyle i}$ ${\displaystyle e_{i}}$ ${\displaystyle 1}$ ${\displaystyle 0}$ ${\displaystyle I}$ ${\displaystyle (F_{i}|i\in I)}$ ${\displaystyle \bigoplus _{i\in I}F_{i}}$ ${\displaystyle F_{i}}$ ${\displaystyle R^{(I)}}$
4. The product of a family of free modules is generally not free. For example, it is not free. ${\displaystyle \mathbb {Z} ^{\mathbb {N} }}$
5. The polynomial ring above the ring is a free module with a basis . ${\displaystyle \textstyle R[X]}$ ${\displaystyle R}$ ${\displaystyle (X^{i}|i\in \mathbb {N} )}$
6. The set of positive rational numbers is a commutative group with respect to multiplication. Because of the unique prime factorization, each can be clearly written with prime numbers . So it is a free Abelian group with a countable base. ${\displaystyle \mathbb {Q} ^{+}}$ ${\displaystyle r\in \mathbb {Q} ^{+}}$ ${\displaystyle r=p_{1}^{z_{1}}\cdots p_{n}^{z_{n}}}$ ${\displaystyle p_{1},\dots ,p_{n}}$ ${\displaystyle \mathbb {Q} ^{+}}$
7. The ring is an inclined body if and only if every module above this ring is free. ${\displaystyle R}$

The rank of a free module

Many of the theorems about bases of vector spaces no longer apply to free modules:

1. If there is a vector space above the body with a base of elements, then every system of free elements is also a generating system, i.e. a base. This generally does not apply to rings: For example, in the module the quantity is free, but no basis. ${\displaystyle \textstyle V}$ ${\displaystyle K}$ ${\displaystyle \textstyle n}$ ${\displaystyle \textstyle n}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle \textstyle \{2\}}$
2. If a vector space is, then every two bases are equally powerful. This still applies to commutative rings. So if the ring is commutative and so is . A short, relatively elementary proof of this can be found in the book by Jens Carsten Jantzen and Joachim Schwermer . About non-commutative rings, the theorem is generally wrong. An example of this is given in the book mentioned. One can therefore not generally define the rank of a free module. Rings in which two bases of a free module are equally thick are called IBN rings. Noether's rings have this property. ${\displaystyle V}$ ${\displaystyle R}$ ${\displaystyle R^{n}\cong R^{m}}$ ${\displaystyle n=m}$
3. It is more general: if it is a homomorphism of rings and is an IBN ring, so too . For example, if there is a ring homomorphism after a Noetherian ring , then it is an IBN ring. ${\displaystyle \rho \colon R\rightarrow S}$ ${\displaystyle S}$ ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle S}$ ${\displaystyle R}$

Properties of free modules

General properties

1. If there is a family of elements from the module , there is exactly one homomorphism with . Thereby a base (in case of doubt the canonical) of . If the family creates the module , there is an epimorphism. Each module is therefore an epimorphic image of a free module. ${\displaystyle (m_{i}|i\in I)}$ ${\displaystyle M}$ ${\displaystyle R^{(I)}=\bigoplus _{i\in I}Re_{i}\rightarrow M}$ ${\displaystyle f(e_{i})=m_{i}}$ ${\displaystyle (e_{i}|i\in I)}$ ${\displaystyle R^{(I)}}$ ${\displaystyle (m_{i}|i\in I)}$ ${\displaystyle M}$ ${\displaystyle f}$
2. If there is a free module and an epimorphism, then a direct summand is in . There is one with . ${\displaystyle F}$ ${\displaystyle f\colon M\rightarrow F}$ ${\displaystyle \operatorname {Kern} (f)}$ ${\displaystyle M}$ ${\displaystyle g\colon F\rightarrow M}$ ${\displaystyle f\circ g=\mathbf {1} _{F}}$
3. The statement 1. can be expressed more generally and at the same time more precisely. The free module and the canonical injective mapping belong to every set . If there is another set and a mapping between the sets, then there is exactly one homomorphism for the family , so that applies. That is, the following diagram is commutative: ${\displaystyle X}$ ${\displaystyle \mathbf {F} (X):=R^{(X)}}$ ${\displaystyle \Phi (X)\colon X\ni x\mapsto e_{x}\in R^{(X)}}$ ${\displaystyle Y}$ ${\displaystyle \alpha \colon X\rightarrow Y}$ ${\displaystyle (e_{\alpha (x)}|x\in X)}$ ${\displaystyle \mathbf {F} (\alpha )\colon \mathbf {F} (X)\rightarrow \mathbf {F} (Y)}$ ${\displaystyle \mathbf {F} (\alpha )\circ \Phi (X)=\Phi (Y)\circ \alpha }$
   
   Are illustrations, so is . In the language of category theory it can be expressed as follows: is a true functor from the category of sets to the category of free modules. is a functional monomorphism between the identity functor and the functor . ${\displaystyle \alpha \colon X\rightarrow Y\,\beta \colon Y\rightarrow Z}$ ${\displaystyle \mathbf {F} (\alpha \circ \beta )=\mathbf {F} (\alpha )\circ \mathbf {F} (\beta )}$ ${\displaystyle \mathbf {F} }$ ${\displaystyle \Phi }$ ${\displaystyle \mathbf {F} }$
4. As in 3, the free module belongs to each module . This includes the clearly determined epimorphism . For everyone is . It is a functional epimorphism between the functor and the identity functor . ${\displaystyle M}$ ${\displaystyle F(M)=R^{(M)}=\bigoplus _{m\in M}Re_{m}}$ ${\displaystyle \Psi (M):F(M)\ni e_{m}\mapsto m\in M}$ ${\displaystyle \alpha \colon M\rightarrow N}$ ${\displaystyle \Psi (N)\circ F(\alpha )=\alpha \circ \Psi (N)}$ ${\displaystyle \Psi }$ ${\displaystyle \mathbf {F} }$

Free modules over special rings

1. Each sub-module of a free module is free again via main ideal rings .
2. All direct summands of free modules (that is, projective modules ) are free over local rings .

construction

There is a free link module for every quantity and every ring . Its carrier is the set of formal linear combinations of elements, coded roughly as . Addition and scalar multiplication take place point by point: ${\displaystyle S}$ ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle FS}$ ${\displaystyle S}$ ${\displaystyle S}$ ${\displaystyle FS:=\{v\colon S\to R\mid \{s\in S\mid v(s)\neq 0\}{\text{ endlich}}\}}$

	${\displaystyle +}$	${\displaystyle \colon }$	${\displaystyle FS\times FS\to FS}$	${\displaystyle (v+w)(s):=v(s)+w(s)}$
	${\displaystyle \cdot }$	${\displaystyle \colon }$	${\displaystyle R\times FS\to FS}$	${\displaystyle (r\cdot v)(s):=r\cdot v(s)}$

The elements of are not elements of . If one (or even a left-generating with but) so they can be embedded by ${\displaystyle S}$ ${\displaystyle FS}$ ${\displaystyle R}$ ${\displaystyle 1=:e}$ ${\displaystyle e}$ ${\displaystyle \mathbb {Z} e+Re=R}$

	${\displaystyle \eta _{S}}$	${\displaystyle \colon }$	${\displaystyle S\to FS}$	${\displaystyle \eta _{S}(s):}$	${\displaystyle S\to R}$
					${\displaystyle t\mapsto {\begin{cases}e&s=t\\0&{\text{sonst.}}\end{cases}}}$

The free -Rechtsmodul is the free -Linksmodul, with the counter-ring of designated. ${\displaystyle R}$ ${\displaystyle R^{\text{op}}}$ ${\displaystyle R^{\text{op}}}$ ${\displaystyle R}$

Attenuations

The following diagram relates the freedom of a module over a commutative ring with the properties projective , flat and torsion-free : ${\displaystyle M}$ ${\displaystyle A}$

See also

• Base (vector space)
• Projective object

literature

• Tsit-Yuen Lam: Lectures on modules and rings. GTM 189, Springer, 1999, ISBN 0-387-98428-3 .
• Friedrich Kasch: modules and rings. Teubner, Stuttgart 1977, ISBN 3-519-02211-7 .
• Robert Wisbauer: Fundamentals of module and ring theory. Reinhard Fischer, Munich 1988, ISBN 3-88927-044-1 .

Individual evidence

1. ↑ Tsit-Yuen Lam: Lectures on modules and rings. GTM 189, Springer, 1999, ISBN 0-387-98428-3 , p. 22 f.
2. ↑ Jens Carsten Jantzen, Joachim Schwermer: Algebra , Springer, 2006, ISBN 3-540-21380-5 , doi: 10.1007 / 3-540-29287-X , page 194
3. ↑ See also the article en: Invariant basis number