A theorem of Kaplansky revisited

TL;DR

This paper provides a new proof of Kaplansky's theorem on matrix semigroup triangularizability, extends the theorem, and explores its implications for groups of unipotent matrices over division rings.

Contribution

It offers a simplified proof of Kaplansky's theorem, extends it to new contexts, and connects it with Kolchin's theorem over division rings.

Findings

Unified theorem of Kolchin and Levitzki on triangularizability

Extensions of Kaplansky's theorem to new algebraic structures

Counterparts of Kolchin's theorem over division rings

Abstract

We present a new and simple proof of a theorem due to Kaplansky which unifies theorems of Kolchin and Levitzki on triangularizability of semigroups of matrices. We also give two different extensions of the theorem. As a consequence, we prove the counterpart of Kolchin's Theorem for finite groups of unipotent matrices over division rings. We also show that the counterpart of Kolchin's Theorem over division rings of characteristic zero implies that of Kaplansky's Theorem over such division rings.