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Absolute stable rank and Witt cancellation for noncommutative rings. (English) Zbl 0639.16015
Bass introduced the notion of “stable range” conditions on a ring R in order to characterize those integers n for which every matrix in GL${}\sb n$(R) can be row reduced to a matrix with the same last row and column as the identity matrix. Similar questions concerning orthogonal groups over commutative rings led M. R. Stein to introduce “absolute stable range” conditions for commutative rings. The authors of this paper take up the question of absolute stable range and its connection with cancellation of quadratic forms over non-commutative rings. They provide a concise survey of previous results and their interconnections (including some examples of rings whose stable ranges and absolute stable ranges differ) and prove several new and interesting theorems.
Reviewer: M.R.Stein

16E20Grothendieck groups and $K$-theory of noncommutative rings
20G35Linear algebraic groups over adèles and other rings and schemes
11E16General binary quadratic forms
Full Text: DOI EuDML
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