SL_⁎ over local and adèle rings: ⁎-Euclideanity and Bruhat generators

Let (R,⁎) be a ring with involution and let A=M(n,R) be the matrix ring endowed with the ⁎-transpose involution. We study SL_⁎(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 ⁎-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 whether a commutative adèlic ring R leads towards the ring A being ⁎-Euclidean; while the non-commutative adèlic quaternions are such that A is ⁎-Euclidean and SL_⁎ is generated by its Bruhat elements if and only if the characteristic is 2.