Absolute stable rank and Witt cancellation for noncommutative rings (Q1100258)

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\({}_ n\)(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.