To the best of my knowledge, Bass defined the notion of stable range. Then somewhere it was shown that the condition on unimodular sequences defining the stable range holds for $N\geq n$ if it holds for $n$. Can somebody please tell me where this theorem is proven and where the notion of stable rank first appears?
Thank you in advance!
On
According to Lam (who would know well: Bass was his advisor)
The concept of stable range was initiated by H. Bass in his investigation of the stability properties of the general linear group in algebraic K-theory (H. Bass: K-theory and stable algebra, Publ. IHES 22(1964), 5–60).
This quotation appears in his writeup Lam, T. Y. "A crash course on stable range, cancellation, substitution and exchange." Journal of Algebra and Its Applications 3.03 (2004): 301-343.
I could not spot the proposition in this 1964 paper, but I believe it is Proposition 1.3 in Lam's exposition.
I'm no expert on this, or its history, but the specific result you mention (that if the condition holds for some $n$ then it holds for all larger $N$) is Theorem 1 of
Vaserstein, L. N., Stable rank of rings and dimensionality of topological spaces, Funct. Anal. Appl. 5, 102-110 (1971); translation from Funkts. Anal. Prilozh. 5, No. 2, 17-27 (1971).
which might well be its first appearance.
The link is to the English translation of a paper first published in Russian.