Reverse decomposition of unipotents over noncommutative rings I: General linear groups

Abstract

Recently is has been proved that if $\sigma \in G{L}_n(R)$ where $R$ is an commutative ring and $n\ge 3$, then each of the elementary transvections $t_{kl}({\sigma }_{ij})(i\ne j,k\ne l)$ is a product of eight $E_n(R)$-conjugates of $\sigma$ and ${\sigma }^{-1}$. In this article we show that similar results hold true if $R$ is a (noncommutative) von Neumann regular ring, or a Banach algebra, or a ring satisfying a stable range condition, or a ring with Euclidean algorithm, or an almost commutative ring.