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A free object (a free algebra) in the variety of modules over a fixed ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041600/f0416001.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041600/f0416002.png" /> is associative and has a unit, then a free module is a module with a basis, that is, a linearly independent system of generators. The cardinality of a basis of a free module is called its rank. The rank is not always defined uniquely, that is, there are rings over which a free module can have two bases consisting of a different number of elements. This is equivalent to the existence over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041600/f0416003.png" /> of two rectangular matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041600/f0416004.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041600/f0416005.png" /> for which
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A free object (a free algebra) in the variety of modules over a fixed ring $R$. If $R$ is associative and has a unit, then a free module is a module with a basis, that is, a linearly independent system of generators. The cardinality of a basis of a free module is called its rank. The rank is not always defined uniquely, that is, there are rings over which a free module can have two bases consisting of a different number of elements. This is equivalent to the existence over $R$ of two rectangular matrices $A$ and $B$ for which
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041600/f0416006.png" /></td> </tr></table>
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$$AB=I_m,\quad BA=I_n,\quad m\neq n,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041600/f0416007.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041600/f0416008.png" /> denote the unit matrices of orders <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041600/f0416009.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041600/f04160010.png" />, respectively. However, non-uniqueness holds only for finite bases; if the rank of a free module is infinite, then all bases have the same cardinality. In addition, over rings that admit a homomorphism into a skew-field (in particular, over commutative rings), the rank of a free module is always uniquely defined.
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where $I_m$ and $I_n$ denote the unit matrices of orders $m$ and $n$, respectively. However, non-uniqueness holds only for finite bases; if the rank of a free module is infinite, then all bases have the same cardinality. In addition, over rings that admit a homomorphism into a skew-field (in particular, over commutative rings), the rank of a free module is always uniquely defined.
  
A ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041600/f04160011.png" />, considered as a left module over itself, is a free module of rank 1. Every left free module is a direct sum of free modules of rank 1. Every module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041600/f04160012.png" /> is representable as a quotient module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041600/f04160013.png" /> of a free module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041600/f04160014.png" />. The submodule <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041600/f04160015.png" /> is, in turn, representable as a quotient module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041600/f04160016.png" /> of a free module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041600/f04160017.png" />. By continuing this process one obtains the exact sequence
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A ring $R$, considered as a left module over itself, is a free module of rank 1. Every left free module is a direct sum of free modules of rank 1. Every module $M$ is representable as a quotient module $F_0/H_0$ of a free module $F_0$. The submodule $H_0$ is, in turn, representable as a quotient module $F_1/H_1$ of a free module $F_1$. By continuing this process one obtains the exact sequence
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041600/f04160018.png" /></td> </tr></table>
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$$\dotsb\to F_2\to F_1\to F_0\to M\to0,$$
  
which is called the [[Free resolution|free resolution]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041600/f04160019.png" />. Skew-fields can be characterized as rings over which all modules are free. Over a principal ideal domain a submodule of a free module is free. Near to free modules are projective modules and flat modules (cf. [[Projective module|Projective module]]; [[Flat module|Flat module]]).
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which is called the [[Free resolution|free resolution]] of $M$. Skew-fields can be characterized as rings over which all modules are free. Over a principal ideal domain a submodule of a free module is free. Near to free modules are projective modules and flat modules (cf. [[Projective module|Projective module]]; [[Flat module|Flat module]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.M. Cohn,  "Free rings and their relations" , Acad. Press  (1971)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. MacLane,  "Homology" , Springer  (1963)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.M. Cohn,  "Free rings and their relations" , Acad. Press  (1971)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. MacLane,  "Homology" , Springer  (1963)</TD></TR></table>

Latest revision as of 12:24, 14 February 2020

A free object (a free algebra) in the variety of modules over a fixed ring $R$. If $R$ is associative and has a unit, then a free module is a module with a basis, that is, a linearly independent system of generators. The cardinality of a basis of a free module is called its rank. The rank is not always defined uniquely, that is, there are rings over which a free module can have two bases consisting of a different number of elements. This is equivalent to the existence over $R$ of two rectangular matrices $A$ and $B$ for which

$$AB=I_m,\quad BA=I_n,\quad m\neq n,$$

where $I_m$ and $I_n$ denote the unit matrices of orders $m$ and $n$, respectively. However, non-uniqueness holds only for finite bases; if the rank of a free module is infinite, then all bases have the same cardinality. In addition, over rings that admit a homomorphism into a skew-field (in particular, over commutative rings), the rank of a free module is always uniquely defined.

A ring $R$, considered as a left module over itself, is a free module of rank 1. Every left free module is a direct sum of free modules of rank 1. Every module $M$ is representable as a quotient module $F_0/H_0$ of a free module $F_0$. The submodule $H_0$ is, in turn, representable as a quotient module $F_1/H_1$ of a free module $F_1$. By continuing this process one obtains the exact sequence

$$\dotsb\to F_2\to F_1\to F_0\to M\to0,$$

which is called the free resolution of $M$. Skew-fields can be characterized as rings over which all modules are free. Over a principal ideal domain a submodule of a free module is free. Near to free modules are projective modules and flat modules (cf. Projective module; Flat module).

References

[1] P.M. Cohn, "Free rings and their relations" , Acad. Press (1971)
[2] S. MacLane, "Homology" , Springer (1963)
How to Cite This Entry:
Free module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Free_module&oldid=13029
This article was adapted from an original article by V.E. Govorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article