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

Luis Gutiérrez Frez, Luis Lomelí, José Pantoja

Research output: Contribution to journalArticlepeer-review

Abstract

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.

Original languageEnglish
Pages (from-to)1-28
Number of pages28
JournalJournal of Algebra
Volume589
DOIs
StatePublished - 1 Jan 2022
Externally publishedYes

Keywords

  • Bruhat generators
  • SL groups
  • ⁎-Euclidean rings

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