## 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 language | English |
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Pages (from-to) | 1-28 |

Number of pages | 28 |

Journal | Journal of Algebra |

Volume | 589 |

DOIs | |

State | Published - 1 Jan 2022 |

Externally published | Yes |

## Keywords

- Bruhat generators
- SL groups
- ⁎-Euclidean rings