An extension of a theorem of Kaplansky

TL;DR

This paper provides a new proof of Kaplansky's theorem on triangularizability of matrix semigroups with singleton spectra, extends it to division rings, and generalizes it over arbitrary fields.

Contribution

It introduces a simplified proof of Kaplansky's theorem in characteristic zero and extends the theorem to division rings and general fields.

Findings

New simple proof of Kaplansky's Theorem in characteristic zero

Extension of the theorem to division rings of characteristic zero

Generalization of the theorem over arbitrary fields

Abstract

A theorem of Kaplansky asserts that a semigroup of matrices with entries from a field whose members all have singleton spectra is triangularizable. Indeed, Kaplansky's Theorem unifies well-known theorems of Kolchin and Levitzki on simultaneous triangularizability of semigroups of unipotent and nilpotent matrices, respectively. First, we present a new and simple proof of Kaplansky's Theorem over fields of characteristic zero. Next, we show that this proof can be adjusted to show that the counterpart of Kolchin's Theorem over division rings of characteristic zero implies that of Kaplansky's Theorem over such division rings. Also, we give a generalization of Kaplansky's Theorem over general fields. We show that this extension of Kaplansky's Theorem holds over a division ring $\Delta$ provided the counterpart of Kaplansky's Theorem holds over $\Delta$.