Division Rings with Ranks

TL;DR

This paper investigates division rings with model-theoretic ranks, establishing their finite-dimensionality over centers, and classifies interpretable division rings in o-minimal structures as algebraically closed, real closed, or quaternions.

Contribution

It proves superrosy division rings are centrally finite and classifies interpretable division rings in o-minimal structures.

Findings

Superrosy division rings are centrally finite.

Division rings of burden n have dimension at most n over their center.

Interpretable division rings in o-minimal structures are algebraically closed, real closed, or quaternions.

Abstract

Any superrosy division ring (i.e. a division ring equipped with an abstract notion of rank) is shown to be centrally finite. Furthermore, division rings satisfying a generalized chain condition on definable subgroups are studied. In particular, a division ring of burden n has dimension at most n over its center, and any definable group of definable automorphisms of a field of burden n has size at most n. Additionally, interpretable division rings in o-minimal structures are shown to be algebraically closed, real closed or the quaternions over a real closed field.