On division subrings normalized by almost subnormal subgroups in division rings

Triangulaizability of Algebras Over Division Rings

On nest modules of matrices over division rings

Let $ m , n in mathbb{N}$, $D$ be a division ring, and $M_{m times n}(D)$ denote the bimodule of all $m times n$ matrices with entries from $D$. First, we characterize one-sided submodules of $M_{m times n}(D)$ in terms of left row reduced echelon or right column reduced echelon matrices with entries from $D$. Next, we introduce the notion of a nest module of matrices with entries from $D$. We ...

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...

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...

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...