  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.
