Division Rings with Ranks

Nadja Hempel (1) , Daniel Palacín (2)

1 AGL - Algèbre, géométrie, logique
2 Institut für Mathematische Logik und Grundlagenforschung

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.

Mots clés: division rings; model theory; superrosy; burden

Domaines: Logique [math.LO]; Anneaux et algèbres [math.RA]

Nadja Hempel, Daniel Palacín. Division Rings with Ranks. 2016. ⟨hal-01315131⟩