View source for Stable rank

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" />.

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.

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" />.

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]]].

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.

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.

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" />.

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).

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.

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).

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.

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 <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]]].

====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>