Rings with transfinite left division algorithm

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...

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...

Transfinite Braids and Left Distributive Operations

We complete Artin’s braid group B∞ with some limit points (with respect to a natural topology), thus obtaining an extended monoid where new left self-distributive operations are defined. This construction provides an effective realization for some free algebraic system involving a left distributive operation and a compatible associative product. AMS Subject classification: 20F36, 20N02 Here we ...

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...