A note on algebraic commutators in division rings with uncountable center

Hoang Minh Thu, Vo

doi:10.22060/ajmc.2025.23898.1324

A note on algebraic commutators in division rings with uncountable center

Articles in Press

Document Type : Original Article

Author

Vo Hoang Minh Thu

Faculty of Mathematics and Computer Science, University of Sci- ence, Ho Chi Minh City, Vietnam Vietnam National University, Ho Chi Minh City, Vietnam

10.22060/ajmc.2025.23898.1324

Abstract

Let $D$ be a division ring with uncountable center $C$. Suppose that $ K $ is a sub-division ring of $ D $ containing $C$ and that $ a \in D \setminus C $. The purpose of this paper is to prove that if either $ axa^{-1}x^{-1} $ or $ xy - yx $ is right algebraic over $ K $ for all $ x, y \in D \setminus \{0\} $, then $ D $ is also right algebraic over $ K $. This result provides the affirmative answers to [8, Problems 1 and 5] for division rings with uncountable center.

Keywords

algebraic element

uncountable center

division ring

left algebraic

right algebraic

Subjects