Algebraic commutators with respect to subnormal subgroups in division rings

TL;DR

This paper investigates the structure of division rings by examining algebraic properties of subgroups and commutators over subfields, establishing bounds on the degree of the division ring over its center.

Contribution

It introduces new bounds on the degree of division rings based on algebraic properties of subnormal subgroups and commutators over subfields.

Findings

If a noncentral normal subgroup is left algebraic over K of bounded degree, then [D:F] ≤ d^2.

Algebraicity of additive or multiplicative commutators over F implies [D:F] ≤ d^2.

Results extend understanding of division ring structure via algebraic properties of subgroups.

Abstract

Let $D$ be a division ring and $K$ a subfield of $D$ which is not necessarily contained in the center $F$ of $D$. In this paper, we study the structure of $D$ under the condition of left algebraicity of certain subsets of $D$ over $K$. Among results, it is proved that if $D^*$ contains a noncentral normal subgroup which is left algebraic over $K$ of bounded degree $d$, then $[D:F]\le d^2$. In case $K=F$, the obtained results show that if either all additive commutators or all multiplicative commutators with respect to a noncentral subnormal subgroup of $D^*$ are algebraic of bounded degree $d$ over $F$, then $[D:F]\le d^2$.