  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.
