${\mathrm{SL}}_{\ast }$ over local and adèle rings: $\ast$-euclideanity and Bruhat generators

Abstract

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