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

TL;DR

This paper extends the decomposition of unipotent elements in general linear groups from commutative rings to various classes of noncommutative rings, broadening the understanding of algebraic structures in these contexts.

Contribution

It generalizes the reverse decomposition results of unipotents over commutative rings to several classes of noncommutative rings, including von Neumann regular rings and Banach algebras.

Findings

Decomposition results hold for von Neumann regular rings.

Results apply to Banach algebras and rings with stable range conditions.

Extends known decompositions to almost commutative rings.

Abstract

Recently is has been proved that if $\sigma\in GL_n(R)$ where $R$ is an commutative ring and $n\geq 3$, then each of the elementary transvections $t_{kl}(\sigma_{ij})~(i\neq j,k\neq l)$ 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.