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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s0871301.png" /> be an associative ring with unit element. A sequence of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s0871302.png" /> is called left unimodular if the left ideal generated by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s0871303.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s0871304.png" />, is all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s0871305.png" />.
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The left stable rank of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s0871306.png" /> is the least integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s0871307.png" /> such that for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s0871308.png" /> and left unimodular sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s0871309.png" /> there are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713010.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713011.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713013.png" />, is also left unimodular.
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The right stable rank of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713014.png" /> is defined analogously by replacing left with right everywhere. The left and right stable ranks are equal [[#References|[a1]]], cf. also, e.g., [[#References|[a2]]], §11.3, and both are therefore called the stable rank of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713015.png" />, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713016.png" />.
+
Let  $  R $
 +
be an associative ring with unit element. A sequence of elements  $  ( a _ {1} \dots a _ {n} ) $
 +
is called left unimodular if the left ideal generated by the  $  a _ {i} $,  
 +
$  i= 1 \dots n $,  
 +
is all of $  R $.
  
By writing a left unimodular sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713017.png" /> as a column, there is a natural left action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713018.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713019.png" />, the set of all left unimodular sequences of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713020.png" />. The general linear rank, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713021.png" />, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713022.png" /> is the least integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713023.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713024.png" /> acts transitively on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713025.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713026.png" />. This is equivalent to the property that all right stably-free modules of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713027.png" /> are free, [[#References|[a2]]].
+
The left stable rank of $  R $
 +
is the least integer $  n $
 +
such that for each  $  m> n $
 +
and left unimodular sequence  $  ( a _ {1} \dots a _ {m} ) $
 +
there are  $  r _ {1} \dots r _ {m-} 1 $
 +
such that $  ( a _ {1}  ^  \prime  \dots a _ {m-} 1  ^  \prime  ) $
 +
with  $  a _ {i}  ^  \prime  = a _ {i} + r _ {i} a _ {m} $,
 +
$  i= 1 \dots m- 1 $,  
 +
is also left unimodular.
  
Recall that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713028.png" /> is stably free if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713029.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713030.png" />; the rank of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713031.png" /> is then defined as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713032.png" />. This is well-defined if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713033.png" /> has the invariant basis property (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713034.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713035.png" />). This property holds, e.g., if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713036.png" /> is commutative or right Noetherian.
+
The right stable rank of $  R $
 +
is defined analogously by replacing left with right everywhere. The left and right stable ranks are equal [[#References|[a1]]], cf. also, e.g., [[#References|[a2]]], §11.3, and both are therefore called the stable rank of  $  R $,
 +
denoted by  $  \textrm{ st.r. } ( R) $.
  
One has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713037.png" />, so that any stably-free module of rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713038.png" /> is free.
+
By writing a left unimodular sequence  $  ( a _ {1} \dots a _ {n} ) $
 +
as a column, there is a natural left action of  $  \mathop{\rm GL} _ {n} ( R ) $
 +
on  $  U _ {c} ( n, R ) $,
 +
the set of all left unimodular sequences of length  $  n $.  
 +
The general linear rank,  $  \mathop{\rm glr} ( R) $,
 +
of  $  R $
 +
is the least integer  $  n $
 +
such that  $  \mathop{\rm GL} _ {m} ( R) $
 +
acts transitively on  $  U _ {c} ( m, R) $
 +
for all  $  m> n $.  
 +
This is equivalent to the property that all right stably-free modules of rank $  \geq  n $
 +
are free, [[#References|[a2]]].
  
For a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713039.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713040.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713041.png" />.
+
Recall that  $  P $
 +
is [[Stably free module|stably free]] if  $  P\oplus R  ^ {n} \simeq R  ^ {m} $
 +
for some  $  n, m $;
 +
the rank of  $  P $
 +
is then defined as  $  m- n $.  
 +
This is well-defined if  $  R $
 +
has the invariant basis property (i.e. $  R  ^ {n} \simeq R  ^ {m} $
 +
if and only if  $  n= m $).  
 +
This property holds, e.g., if  $  R $
 +
is commutative or right Noetherian.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713042.png" /> be a field of transcendence degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713043.png" /> over its prime subfield <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713044.png" />. The Kronecker dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713045.png" /> is then defined as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713046.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713047.png" /> and as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713048.png" /> otherwise. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713049.png" /> Kronecker dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713050.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713051.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713052.png" /> is commutative of Krull dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713053.png" /> (cf. also [[Dimension|Dimension]] of an associative ring), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713054.png" /> (Bass' theorem).
+
One has  $  \mathop{\rm glr} ( R) \leq  \textrm{ st.r. } ( R) $,  
 +
so that any stably-free module of rank  $  \geq  \textrm{ st.r. } ( R) $
 +
is free.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713055.png" /> be a topological space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713056.png" /> a metric space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713057.png" /> a continuous mapping. A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713058.png" /> is a stable value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713059.png" /> if it is in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713060.png" /> and if there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713061.png" /> such that for every continuous mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713062.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713063.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713064.png" /> it is still true that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713065.png" />. The mapping dimension of a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713067.png" />, is the largest integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713068.png" /> for which there exists a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713069.png" /> for which the origin is a stable value. (If no such <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713070.png" /> exists, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713071.png" /> is set equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713072.png" />.) For nice spaces, e.g., metrizable, separable, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713073.png" />, this concept of dimension coincides with other notions of dimension, such as inductive dimension, [[#References|[a5]]], Chapt. VI, §1 (cf. [[Dimension theory|Dimension theory]]). It always coincides with the notion of dimension defined by essential mappings (cf. [[Dimension theory|Dimension theory]]), [[#References|[a5]]], Chapt. VI, §3.
+
For a field  $  k $
 +
one has  $  \mathop{\rm glr} ( k[ X _ {1} \dots X _ {n} ])= 1 $
 +
for all  $  n $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713074.png" /> be the ring of real-valued continuous functions on a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713075.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713076.png" /> the subring of bounded functions. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713077.png" /> (Vaserstein's theorem).
+
Let $  k $
 +
be a field of transcendence degree  $  t $
 +
over its prime subfield  $  k _ {0} $.  
 +
The Kronecker dimension of $  k $
 +
is then defined as  $  t+ 1 $
 +
if  $  \mathop{\rm char}  k= 0 $
 +
and as  $  t $
 +
otherwise. For  $  n\leq  $
 +
Kronecker dimension of  $  k $,
 +
$  \textrm{ st.r. } ( k[ X _ {1} \dots X _ {n} ]) = n+ 1 $.  
 +
If  $  R $
 +
is commutative of Krull dimension  $  m< \infty $(
 +
cf. also [[Dimension|Dimension]] of an associative ring), then  $  \textrm{ st.r. } ( R[ X _ {1} \dots X _ {n} ]) \leq  m+ n+ 1 $(
 +
Bass' theorem).
  
Both Bass' and Vaserstein's theorem indicate that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713078.png" /> is a good dimension concept for rings.
+
Let  $  X $
 +
be a topological space,  $  Y $
 +
a metric space and  $  f:  X \rightarrow Y $
 +
a continuous mapping. A point  $  y \in Y $
 +
is a stable value of  $  f $
 +
if it is in  $  f( X) $
 +
and if there is an  $  \epsilon $
 +
such that for every continuous mapping  $  g:  X \rightarrow Y $
 +
with  $  \| f( x)- g( x) \| < \epsilon $
 +
for all  $  x \in X $
 +
it is still true that  $  y \in g( X) $.
 +
The mapping dimension of a topological space  $  X $,
 +
$  d( X) $,
 +
is the largest integer  $  d $
 +
for which there exists a mapping  $  X \rightarrow \mathbf R  ^ {d} $
 +
for which the origin is a stable value. (If no such  $  d $
 +
exists,  $  d( X) $
 +
is set equal to  $  \infty $.)
 +
For nice spaces, e.g., metrizable, separable,  $  X $,
 +
this concept of dimension coincides with other notions of dimension, such as inductive dimension, [[#References|[a5]]], Chapt. VI, §1 (cf. [[Dimension theory|Dimension theory]]). It always coincides with the notion of dimension defined by essential mappings (cf. [[Dimension theory|Dimension theory]]), [[#References|[a5]]], Chapt. VI, §3.
  
More generally, the stable rank is defined for subrings and ideals of an associative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713079.png" /> with unit.
+
Let  $  C( X) $
 +
be the ring of real-valued continuous functions on a topological space  $  X $
 +
and $  C _ {b} ( X) \subset  C( X) $
 +
the subring of bounded functions. Then  $  \textrm{ st.r.  } ( C( X)) = \textrm{ st.r. } ( C _ {b} ( X))= d( X)+ 1 $(
 +
Vaserstein's theorem).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713080.png" /> be an associative ring with unit, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713081.png" /> be a subring (possibly without unit) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713082.png" />. A sequence of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713083.png" /> is left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713085.png" />-unimodular if it is left unimodular (in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713086.png" />) and, moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713087.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713088.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713089.png" />. The stable rank of the subring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713090.png" /> is the least number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713091.png" /> such that for each left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713092.png" />-unimodular sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713093.png" /> of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713094.png" /> there are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713095.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713096.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713097.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713098.png" />, is a left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s08713099.png" />-unimodular sequence of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s087130100.png" />. (Such a property is referred to as a stable range condition, cf., e.g., [[#References|[a4]]]). The stable rank of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s087130101.png" /> does not depend on the ambient ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s087130102.png" />. Again it is true that the notion of stable rank is left/right symmetric, [[#References|[a1]]].
+
Both Bass' and Vaserstein's theorem indicate that  $  \textrm{ st.r. } ( R) - 1 $
 +
is a good dimension concept for rings.
 +
 
 +
More generally, the stable rank is defined for subrings and ideals of an associative ring  $  R $
 +
with unit.
 +
 
 +
Let  $  R $
 +
be an associative ring with unit, and let $  \mathfrak q $
 +
be a subring (possibly without unit) of $  R $.  
 +
A sequence of elements $  ( a _ {1} \dots a _ {n} ) $
 +
is left $  \mathfrak q $-
 +
unimodular if it is left unimodular (in $  R $)  
 +
and, moreover, $  a _ {1} - 1 \in \mathfrak q $,  
 +
$  a _ {i} \in \mathfrak q $,  
 +
$  i= 2 \dots n $.  
 +
The stable rank of the subring $  \mathfrak q $
 +
is the least number $  n $
 +
such that for each left $  \mathfrak q $-
 +
unimodular sequence $  ( a _ {1} \dots a _ {m} ) $
 +
of length $  m> n $
 +
there are $  q _ {i} \in \mathfrak q $,  
 +
$  i= 1 \dots m- 1 $,  
 +
such that $  ( a _ {1}  ^  \prime  \dots a _ {m-} 1  ^  \prime  ) $,  
 +
with $  a _ {i}  ^  \prime  = a _ {i} + q _ {i} a _ {m} $,  
 +
is a left $  \mathfrak q $-
 +
unimodular sequence of length $  m- 1 $.  
 +
(Such a property is referred to as a stable range condition, cf., e.g., [[#References|[a4]]]). The stable rank of $  \mathfrak q $
 +
does not depend on the ambient ring $  R $.  
 +
Again it is true that the notion of stable rank is left/right symmetric, [[#References|[a1]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> L.N. Vaserstein, "Stable ranks of rings and dimensionality of topological spaces" ''Funct. Anal. Appl.'' , '''5''' (1971) pp. 102–110 ''Funkts. Anal. i Prilozhen.'' , '''5''' : 2 (1970) pp. 17–27</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J.C. McConnell, J.C. Robson, "Noncommutative Noetherian rings" , Wiley (1987) {{MR|934572}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> A.J. Hahn, O.T. O'Meara, "The classical groups and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s087130103.png" />-theory" , Springer (1981) pp. §4.1 {{MR|1007302}} {{MR|0842441}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> H. Bass, "Algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s087130104.png" />-theory" , Benjamin (1968) pp. Chapt. V, §3 {{MR|249491}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> W. Hurevicz, G. Wallman, "Dimension theory" , Princeton Univ. Press (1948)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> L.N. Vaserstein, "Stable ranks of rings and dimensionality of topological spaces" ''Funct. Anal. Appl.'' , '''5''' (1971) pp. 102–110 ''Funkts. Anal. i Prilozhen.'' , '''5''' : 2 (1970) pp. 17–27</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J.C. McConnell, J.C. Robson, "Noncommutative Noetherian rings" , Wiley (1987) {{MR|934572}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> A.J. Hahn, O.T. O'Meara, "The classical groups and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s087130103.png" />-theory" , Springer (1981) pp. §4.1 {{MR|1007302}} {{MR|0842441}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> H. Bass, "Algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087130/s087130104.png" />-theory" , Benjamin (1968) pp. Chapt. V, §3 {{MR|249491}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> W. Hurevicz, G. Wallman, "Dimension theory" , Princeton Univ. Press (1948)</TD></TR></table>

Latest revision as of 08:22, 6 June 2020


Let $ R $ be an associative ring with unit element. A sequence of elements $ ( a _ {1} \dots a _ {n} ) $ is called left unimodular if the left ideal generated by the $ a _ {i} $, $ i= 1 \dots n $, is all of $ R $.

The left stable rank of $ R $ is the least integer $ n $ such that for each $ m> n $ and left unimodular sequence $ ( a _ {1} \dots a _ {m} ) $ there are $ r _ {1} \dots r _ {m-} 1 $ such that $ ( a _ {1} ^ \prime \dots a _ {m-} 1 ^ \prime ) $ with $ a _ {i} ^ \prime = a _ {i} + r _ {i} a _ {m} $, $ i= 1 \dots m- 1 $, is also left unimodular.

The right stable rank of $ R $ is defined analogously by replacing left with right everywhere. The left and right stable ranks are equal [a1], cf. also, e.g., [a2], §11.3, and both are therefore called the stable rank of $ R $, denoted by $ \textrm{ st.r. } ( R) $.

By writing a left unimodular sequence $ ( a _ {1} \dots a _ {n} ) $ as a column, there is a natural left action of $ \mathop{\rm GL} _ {n} ( R ) $ on $ U _ {c} ( n, R ) $, the set of all left unimodular sequences of length $ n $. The general linear rank, $ \mathop{\rm glr} ( R) $, of $ R $ is the least integer $ n $ such that $ \mathop{\rm GL} _ {m} ( R) $ acts transitively on $ U _ {c} ( m, R) $ for all $ m> n $. This is equivalent to the property that all right stably-free modules of rank $ \geq n $ are free, [a2].

Recall that $ P $ is stably free if $ P\oplus R ^ {n} \simeq R ^ {m} $ for some $ n, m $; the rank of $ P $ is then defined as $ m- n $. This is well-defined if $ R $ has the invariant basis property (i.e. $ R ^ {n} \simeq R ^ {m} $ if and only if $ n= m $). This property holds, e.g., if $ R $ is commutative or right Noetherian.

One has $ \mathop{\rm glr} ( R) \leq \textrm{ st.r. } ( R) $, so that any stably-free module of rank $ \geq \textrm{ st.r. } ( R) $ is free.

For a field $ k $ one has $ \mathop{\rm glr} ( k[ X _ {1} \dots X _ {n} ])= 1 $ for all $ n $.

Let $ k $ be a field of transcendence degree $ t $ over its prime subfield $ k _ {0} $. The Kronecker dimension of $ k $ is then defined as $ t+ 1 $ if $ \mathop{\rm char} k= 0 $ and as $ t $ otherwise. For $ n\leq $ Kronecker dimension of $ k $, $ \textrm{ st.r. } ( k[ X _ {1} \dots X _ {n} ]) = n+ 1 $. If $ R $ is commutative of Krull dimension $ m< \infty $( cf. also Dimension of an associative ring), then $ \textrm{ st.r. } ( R[ X _ {1} \dots X _ {n} ]) \leq m+ n+ 1 $( Bass' theorem).

Let $ X $ be a topological space, $ Y $ a metric space and $ f: X \rightarrow Y $ a continuous mapping. A point $ y \in Y $ is a stable value of $ f $ if it is in $ f( X) $ and if there is an $ \epsilon $ such that for every continuous mapping $ g: X \rightarrow Y $ with $ \| f( x)- g( x) \| < \epsilon $ for all $ x \in X $ it is still true that $ y \in g( X) $. The mapping dimension of a topological space $ X $, $ d( X) $, is the largest integer $ d $ for which there exists a mapping $ X \rightarrow \mathbf R ^ {d} $ for which the origin is a stable value. (If no such $ d $ exists, $ d( X) $ is set equal to $ \infty $.) For nice spaces, e.g., metrizable, separable, $ X $, this concept of dimension coincides with other notions of dimension, such as inductive dimension, [a5], Chapt. VI, §1 (cf. Dimension theory). It always coincides with the notion of dimension defined by essential mappings (cf. Dimension theory), [a5], Chapt. VI, §3.

Let $ C( X) $ be the ring of real-valued continuous functions on a topological space $ X $ and $ C _ {b} ( X) \subset C( X) $ the subring of bounded functions. Then $ \textrm{ st.r. } ( C( X)) = \textrm{ st.r. } ( C _ {b} ( X))= d( X)+ 1 $( Vaserstein's theorem).

Both Bass' and Vaserstein's theorem indicate that $ \textrm{ st.r. } ( R) - 1 $ is a good dimension concept for rings.

More generally, the stable rank is defined for subrings and ideals of an associative ring $ R $ with unit.

Let $ R $ be an associative ring with unit, and let $ \mathfrak q $ be a subring (possibly without unit) of $ R $. A sequence of elements $ ( a _ {1} \dots a _ {n} ) $ is left $ \mathfrak q $- unimodular if it is left unimodular (in $ R $) and, moreover, $ a _ {1} - 1 \in \mathfrak q $, $ a _ {i} \in \mathfrak q $, $ i= 2 \dots n $. The stable rank of the subring $ \mathfrak q $ is the least number $ n $ such that for each left $ \mathfrak q $- unimodular sequence $ ( a _ {1} \dots a _ {m} ) $ of length $ m> n $ there are $ q _ {i} \in \mathfrak q $, $ i= 1 \dots m- 1 $, such that $ ( a _ {1} ^ \prime \dots a _ {m-} 1 ^ \prime ) $, with $ a _ {i} ^ \prime = a _ {i} + q _ {i} a _ {m} $, is a left $ \mathfrak q $- unimodular sequence of length $ m- 1 $. (Such a property is referred to as a stable range condition, cf., e.g., [a4]). The stable rank of $ \mathfrak q $ does not depend on the ambient ring $ R $. Again it is true that the notion of stable rank is left/right symmetric, [a1].

References

[a1] L.N. Vaserstein, "Stable ranks of rings and dimensionality of topological spaces" Funct. Anal. Appl. , 5 (1971) pp. 102–110 Funkts. Anal. i Prilozhen. , 5 : 2 (1970) pp. 17–27
[a2] J.C. McConnell, J.C. Robson, "Noncommutative Noetherian rings" , Wiley (1987) MR934572
[a3] A.J. Hahn, O.T. O'Meara, "The classical groups and -theory" , Springer (1981) pp. §4.1 MR1007302 MR0842441
[a4] H. Bass, "Algebraic -theory" , Benjamin (1968) pp. Chapt. V, §3 MR249491
[a5] W. Hurevicz, G. Wallman, "Dimension theory" , Princeton Univ. Press (1948)
How to Cite This Entry:
Stable rank. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stable_rank&oldid=24168