# level of rings and stable range of rings

The level $s(A)$ of a ring $A$ with unity $1$ is the smallest natural number $s$ such that $-1$ is a sum of $s$ squares in $A$. (If $-1$ is not a sum of squares in $A$, we say that $s(A) =\infty$.). The concept was initially introduced for fields by E.Artin and O. Schreier but later studied for commutative rings mainly by Dai, Lam and Peng and for non-commutative rings by Leep and Lewis and several others.

There is another concept well studied in ring theory, viz, Stable range of ring. The concept was introduced by H.Bass. The definition is as follows (taken from Lam's paper A CRASH COURSE ON STABLE RANGE, CANCELLATION, SUBSTITUTION AND EXCHANGE).

Definition 1.1. A sequence ${a_1, . . . , a_n}$ in a ring $R$ is said to be left unimodular if $R_{a_1} + ... + R_{a_n} = R.$ In case $n \geq 2$, such a sequence is said to be reducible if there exist $r_1, . . . , r_{n-1} \in R$ such that $R_{(a_1+r_1a_n)}+...+R_{(a_{n-1}+r_{n-1}a_n)} = R$.

Definition 1.2. A ring R is said to have left stable range $\leq n$ if every left unimodular sequence of length $> n$ is reducible. The smallest such $n$ is said to be the left stable range of $R$; we write simply $sr(R) = n.$ (If no such n exists, we say $sr(R) = \infty$.)

I want to know if there is any kind of relation between these two concepts. I have google searched but could not find any paper relating the two concepts.

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What makes you think that there is such a connection? – Johannes Hahn May 16 '12 at 12:27
The ring of real numbers and the ring of complex numbers take on the two extreme values for $s(A)$ but both have the same stable range. – Steven Landsburg May 16 '12 at 13:06
I found an old paper 1971 which gives connection between dimension of a topological space and the stable range of the ring of real valued continuous functions. So I thought that these two algeb concepts might also be related in some way. – jjm May 16 '12 at 14:53