We investigate the sufficient conditions and the necessary conditions on an exchange ring R under which R has stable range one. These give nontrivial generalizations of Theorem 3 of V. P. Camillo and H.-P. Yu (1995), Theorem 4.19 of K. R. Goodearl (1979, 1991), Theorem 2 of R. E. Hartwig (1982), and Theorem 9 of H.-P. Yu (1995). 2000 Mathematics Subject Classification. Primary 16E50, 19B10. An ...

We study length-minimizing arcs in sub-Riemannian manifolds (M;E;G) whose metric G is de ned on a rank-two bracket-generating distribution E. It is well known that all length-minimizing arcs are extremals, and that these extremals are either \normal" or \abnormal." Normal extremals are locally optimal, in the sense that every su ciently short piece of such an extremal is a minimizer. The questi...