On normal subgroups of division rings which are radical over a proper division subring

TL;DR

This paper introduces Kurosh elements in division rings, generalizing previous results by Faith and Herstein, to explore the structure of normal subgroups that are radical over proper division subrings.

Contribution

It defines Kurosh elements in division rings and extends earlier theorems by Faith and Herstein to a broader context.

Findings

Introduction of Kurosh elements in division rings

Generalization of Faith's result on normal subgroups

Extension of Herstein's theorem

Abstract

We introduce Kurosh elements in division rings based on the idea of a conjecture of Kurosh. Using this, we generalize a result of Faith in [3] and of Herstein in [6].