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, for a division ring D of characteristic zero whose center intersects its multiplicative commutator group in a finite group, we prove that the counterpart of Kolchin’s Theorem over D implies that of Kaplansky’s Theorem over D. Next, we note that this proof, when adjusted in the setting of fields, provides a new and simple proof of Kaplansky’s Theorem over fields of characteristic zero. We show that if Kaplansky’s Theorem holds over a division ring D, which is for instance the case over general fields, then a generalization of Kaplansky’s Theorem holds over D, and in particular over general fields.
Kaplansky's conjectures on group rings
- K[G] does not contain any non-trivial units – if ab = 1 in K[G], then a = k.g for some k in K and g in G.
- K[G] does not contain any non-trivial idempotents – if a2 = a, then a = 1 or a = 0.
Kaplansky's conjecture on Banach algebras
Kaplansky conjecture on Quadratic Forms
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