# A note on multiplicative commutators of division rings

## TL;DR

This paper provides a counterexample in division ring theory, showing that the multiplicative commutator subgroup may not generate the entire division ring as a vector space over its center, challenging a previous conjecture.

## Contribution

It presents the first known example disproving the conjecture that the multiplicative commutator subgroup always generates the division ring as a vector space.

## Key findings

- Counterexample division ring where commutators do not generate the entire ring
- Disproof of the conjecture on vector space generation by multiplicative commutators
- Implication for understanding the structure of division rings

## Abstract

We give an example of a division ring $D$ whose multiplicative commutator subgroup does not generate $D$ as a vector space over its centre, thus disproving the conjecture posed in the paper "Vector space generated by the multiplicative commutators of a division ring, J. Algebra Appl. 12 (2013), no. 8".