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The rank of a left module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077470/r0774701.png" /> over a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077470/r0774702.png" /> imbeddable in a skew-field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077470/r0774703.png" /> is the dimension of the tensor product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077470/r0774704.png" />, regarded as a vector space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077470/r0774705.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077470/r0774706.png" />, the ring of integers, the definition coincides with the usual definition of the rank of an Abelian group (cf. [[Rank of a group|Rank of a group]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077470/r0774707.png" /> is a flat <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077470/r0774708.png" />-module (say, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077470/r0774709.png" /> is the skew-field of fractions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077470/r07747010.png" />, cf. [[Flat module|Flat module]]), then the ranks of the modules in an exact sequence
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<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/r/r077/r077470/r07747011.png" /></td> </tr></table>
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The rank of a left module  $  M $
 +
over a ring  $  R $
 +
imbeddable in a skew-field  $  k $
 +
is the dimension of the tensor product  $  k \otimes _ {R} M $,
 +
regarded as a vector space over  $  k $.
 +
If  $  R = \mathbf Z $,
 +
the ring of integers, the definition coincides with the usual definition of the rank of an Abelian group (cf. [[Rank of a group|Rank of a group]]). If  $  k $
 +
is a flat  $  R $-
 +
module (say,  $  k $
 +
is the skew-field of fractions of  $  R $,
 +
cf. [[Flat module|Flat module]]), then the ranks of the modules in an exact sequence
 +
 
 +
$$
 +
0  \rightarrow  M  ^  \prime  \rightarrow  M  \rightarrow  M  ^ {\prime\prime}  \rightarrow  0
 +
$$
  
 
satisfy the equality
 
satisfy the equality
  
<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/r/r077/r077470/r07747012.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm rk}  M = \mathop{\rm rk}  M  ^  \prime  +  \mathop{\rm rk}  M  ^ {\prime\prime} .
 +
$$
  
The rank of a free module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077470/r07747013.png" /> over an arbitrary ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077470/r07747014.png" /> (cf. [[Free module|Free module]]) is defined as the number of its free generators. For rings that can be imbedded into skew-fields this definition coincides with that in 1). In general, the rank of a free module is not uniquely defined. There are rings (called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077470/r07747016.png" />-FI-rings) such that any free module over such a ring with at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077470/r07747017.png" /> free generators has a uniquely-defined rank, while for free modules with more than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077470/r07747018.png" /> generators this property does not hold. A sufficient condition for the rank of a free module over a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077470/r07747019.png" /> to be uniquely defined is the existence of a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077470/r07747020.png" /> into a skew-field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077470/r07747021.png" />. In this case the concept of the rank of a module can be extended to projective modules as follows. The homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077470/r07747022.png" /> induces a homomorphism of the groups of projective classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077470/r07747023.png" />, and the rank of a projective module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077470/r07747024.png" /> is by definition the image of a representative of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077470/r07747025.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077470/r07747026.png" />. Such a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077470/r07747027.png" /> exists for any commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077470/r07747028.png" />.
+
The rank of a free module $  M $
 +
over an arbitrary ring $  R $(
 +
cf. [[Free module|Free module]]) is defined as the number of its free generators. For rings that can be imbedded into skew-fields this definition coincides with that in 1). In general, the rank of a free module is not uniquely defined. There are rings (called $  n $-
 +
FI-rings) such that any free module over such a ring with at most $  n $
 +
free generators has a uniquely-defined rank, while for free modules with more than $  n $
 +
generators this property does not hold. A sufficient condition for the rank of a free module over a ring $  R $
 +
to be uniquely defined is the existence of a homomorphism $  \phi : R \rightarrow k $
 +
into a skew-field $  k $.  
 +
In this case the concept of the rank of a module can be extended to projective modules as follows. The homomorphism $  \phi $
 +
induces a homomorphism of the groups of projective classes $  \phi  ^ {*} :  K _ {0} R \rightarrow K _ {0} k \approx \mathbf Z $,  
 +
and the rank of a projective module $  P $
 +
is by definition the image of a representative of $  P $
 +
in $  \mathbf Z $.  
 +
Such a homomorphism $  \phi $
 +
exists for any commutative ring $  R $.
  
 
====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">  J.W. Milnor,  "Introduction to algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077470/r07747029.png" />-theory" , Princeton Univ. Press  (1971)</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">  J.W. Milnor,  "Introduction to algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077470/r07747029.png" />-theory" , Princeton Univ. Press  (1971)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The rank of a projective module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077470/r07747030.png" />, as defined here, depends on the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r077/r077470/r07747031.png" />.
+
The rank of a projective module $  P $,  
 +
as defined here, depends on the choice of $  \phi $.

Latest revision as of 08:09, 6 June 2020


The rank of a left module $ M $ over a ring $ R $ imbeddable in a skew-field $ k $ is the dimension of the tensor product $ k \otimes _ {R} M $, regarded as a vector space over $ k $. If $ R = \mathbf Z $, the ring of integers, the definition coincides with the usual definition of the rank of an Abelian group (cf. Rank of a group). If $ k $ is a flat $ R $- module (say, $ k $ is the skew-field of fractions of $ R $, cf. Flat module), then the ranks of the modules in an exact sequence

$$ 0 \rightarrow M ^ \prime \rightarrow M \rightarrow M ^ {\prime\prime} \rightarrow 0 $$

satisfy the equality

$$ \mathop{\rm rk} M = \mathop{\rm rk} M ^ \prime + \mathop{\rm rk} M ^ {\prime\prime} . $$

The rank of a free module $ M $ over an arbitrary ring $ R $( cf. Free module) is defined as the number of its free generators. For rings that can be imbedded into skew-fields this definition coincides with that in 1). In general, the rank of a free module is not uniquely defined. There are rings (called $ n $- FI-rings) such that any free module over such a ring with at most $ n $ free generators has a uniquely-defined rank, while for free modules with more than $ n $ generators this property does not hold. A sufficient condition for the rank of a free module over a ring $ R $ to be uniquely defined is the existence of a homomorphism $ \phi : R \rightarrow k $ into a skew-field $ k $. In this case the concept of the rank of a module can be extended to projective modules as follows. The homomorphism $ \phi $ induces a homomorphism of the groups of projective classes $ \phi ^ {*} : K _ {0} R \rightarrow K _ {0} k \approx \mathbf Z $, and the rank of a projective module $ P $ is by definition the image of a representative of $ P $ in $ \mathbf Z $. Such a homomorphism $ \phi $ exists for any commutative ring $ R $.

References

[1] P.M. Cohn, "Free rings and their relations" , Acad. Press (1971)
[2] J.W. Milnor, "Introduction to algebraic -theory" , Princeton Univ. Press (1971)

Comments

The rank of a projective module $ P $, as defined here, depends on the choice of $ \phi $.

How to Cite This Entry:
Rank of a module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rank_of_a_module&oldid=48433
This article was adapted from an original article by V.E. Govorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article