Stable rank

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)