Stable division rings

Stable division rings

It is shown that a stable division ring with positive characteristic has finite dimension over its centre. This is then extended to simple division rings. Macintyre proved any ω-stable field to be either finite or algebraically closed [7]. This was generalised by Cherlin and Shelah to superstable fields [3]. It follows that a superstable division ring is a field [2]. The result was broadened to...

Triangulaizability of Algebras Over Division Rings

Planar Division Neo-rings

Introduction. The notion of a division ring can be generalized to give a system whose addition is not necessarily associative, but which retains the property of coordinatizing an affine plane. Such a system will be called a planar division neo-ring (PDNR); examples of (infinite) PDNRs which are not division rings are known. If (R, +, •) is a finite power-associative PDNR, then (R, +) is shown t...

Triangularization over finite-dimensional division rings using the reduced trace

In this paper we study triangularization of collections of matrices whose entries come from a finite-dimensional division ring. First, we give a generalization of Guralnick's theorem to the case of finite-dimensional division rings and then we show that in this case the reduced trace function is a suitable alternative for trace function by presenting two triangularization results. The first one...

Supersimple fields and division rings

It is proved that any supersimple field has trivial Brauer group, and more generally that any supersimple division ring is commutative. As prerequisites we prove several results about generic types in groups and fields whose theory is simple.