Division rings with ranks

Abstract

Any superrosy division ring 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, an alternative proof that division rings interpretable in o-minimal structures are algebraically closed, real closed or the quaternions over a real closed field is given.