# On the proof of some theorem on locally nilpotent subgroups in division rings

## TL;DR

This paper provides a new proof for a theorem stating that every locally nilpotent subnormal subgroup in a division ring is central, correcting a previous proof with an identified mistake.

## Contribution

The authors present an alternative proof for a key theorem about locally nilpotent subnormal subgroups in division rings, ensuring mathematical correctness.

## Key findings

- The theorem that such subgroups are central is validated.
- The new proof rectifies previous errors in the literature.
- The result reinforces understanding of subgroup structure in division rings.

## Abstract

In Hai-Thin (2009), there is a theorem, stating that every locally nilpotent subnormal subgroup in a division ring $D$ is central (see Hai-Thin (2009, Th. 2.2)). Unfortunately, there is some mistake in the proof of this theorem. In this note we give the another proof of this theorem.