User bruce a. magurn - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T20:07:41Z http://mathoverflow.net/feeds/user/26144 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/59884/for-which-rings-r-is-sl-nr-generated-by-transvections/106153#106153 Answer by Bruce A. Magurn for For which rings R is SL_n(R) generated by transvections? Bruce A. Magurn 2012-09-02T03:37:49Z 2012-09-02T03:37:49Z <p>Further results are known: L. Vaserstein's paper "SL_2 of Dedekind rings of arithmetic type" proves these rings are generalized euclidean when they have a unit of infinite order. Integral group rings of finite groups are generalized euclidean when the group has no homomorphic image among the generalized quaternion groups of order a multiple of 4, no image among the binary polyhedral groups, and the abelianization of the group has generalized euclidean integral group ring. The finite abelian G with ZG euclidean include the cyclic groups, and Z/2 x Z/2, by the 1984 paper "Generalized euclidean group rings" by Dennis, Magurn &amp; Vaserstein. But ZG is not generalized euclidean when SK_1(Z[G/[G,G]]) is non-vanishing, as it is for Z/4 x Z/2 x Z/2, for instance. So this is a delicate property!</p>