A note On subgroups in a division ring that are left algebraic over a division subring

TL;DR

This paper investigates conditions under which certain subgroups and the entire division ring are left algebraic over a subring, establishing equivalences and bounds related to algebraicity and dimension in division rings.

Contribution

It provides new criteria linking algebraicity of subgroups and the entire division ring, especially under uncountability and containment conditions, and establishes dimension bounds.

Findings

A non-central normal subgroup is left algebraic over K iff D is, given F is uncountable and contained in K.

If the nth derived subgroup is left algebraic of bounded degree, then the division ring's dimension over F is bounded by the square of that degree.

Abstract

Let $D$ be a division ring with center $F$ and $K$ a division subring of $D$. In this paper, we show that a non-central normal subgroup $N$ of the multiplicative group $D^*$ is left algebraic over $K$ if and only if so is $D$ provided $F$ is uncountable and contained in $K$. Also, if $K$ is a field and the $n$-th derived subgroup $D^{(n)}$ of $D^{*}$ is left algebraic of bounded degree $d$ over $K$, then $\dim_FD\le d^2$.