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[edit]
- 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[edit]
Kaplansky conjecture on Quadratic Forms[edit]
References[edit]
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- ^Kaplansky, I. (1951). 'Quadratic forms'. J. Math. Soc. Jpn. 5 (2): 200–207.
- ^Izhboldin, Oleg T. (2001). 'Fields of u-Invariant 9'. Annals of Mathematics. Second Series. 154 (3): 529–587. JSTOR3062141. Zbl0998.11015.
- ^Vishik, Alexander. 'Fields of u-Invariant 2^r + 1'. Algebra, Arithmetic, and Geometry. Progress in Mathematics. 270: 661. doi:10.1007/978-0-8176-4747-6_22.
- H. G. Dales, Automatic continuity: a survey. Bull. London Math. Soc. 10 (1978), no. 2, 129–183.
- W. Lück, L2-Invariants: Theory and Applications to Geometry and K-Theory. Berlin:Springer 2002 ISBN3-540-43566-2
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- M. Puschnigg, The Kadison–Kaplansky conjecture for word-hyperbolic groups. Invent. Math. 149 (2002), no. 1, 153–194.
- H. G. Dales and W. H. Woodin, An introduction to independence for analysts, Cambridge 1987