${\rm SL}_*$ over local and ad\`ele rings: $*$-euclideanity and Bruhat generators

TL;DR

This paper investigates the structure and generation properties of special linear groups with involution over local and ade8lic rings, introducing the concept of *-Euclidean rings and analyzing their implications for Bruhat generation.

Contribution

It introduces the notion of *-local rings, proves that matrix rings over such rings are *-Euclidean, and explores Bruhat generation over various local and ade8lic rings, including non-commutative cases.

Findings

Matrix rings over *-local rings are *-Euclidean.

Over commutative ade8lic rings, the ring is *-Euclidean.

For non-commutative ade8lic quaternions, *-Euclideanity and Bruhat generation depend on characteristic 2.

Abstract

Let $(R,*)$ be a ring with involution and let $A = {\rm M}(n,R)$ be the matrix ring endowed with the $*$-transpose involution. We study ${\rm SL}_*(2,A)$ and the question of Bruhat generation over commutative and non-commutative local and ad\`elic rings $R$. An important tool is the property of a ring being $*$-Euclidean. In this regard, we introduce the notion of a $*$-local ring $R$, prove that $A$ is $*$-Euclidean and explore reduction modulo the Jacobson radical for such rings. Globally, we provide an affirmative answer to the question wether a commutative ad\`elic ring $R$ leads towards the ring $A$ being $*$-Euclidean; while the non-commutative ad\`elic quaternions are such that $A$ is $*$-Euclidean and ${\rm SL}_*$ is generated by its Bruhat elements if and only if the characteristic is $2$.