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**.

## Kaplansky's conjectures on group rings[edit]

*K*be a field, and

*G*a torsion-free group. Kaplansky's zero divisor conjecture states that the group ring

*K*[

*G*] does not contain nontrivial zero divisors, that is, it is a domain. Two related conjectures are:

*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*a*^{2}=*a*, then*a*=*1*or*a*=*0*.

*K*[

*G*] then so does the idempotent conjecture. The latter has been positively solved for an extremely large class of groups, including for example all hyperbolic groups.

## Kaplansky's conjecture on Banach algebras[edit]

*C*(

*X*) (continuous complex-valued functions on

*X*, where

*X*is a compactHausdorff space) into any other Banach algebra, is necessarily continuous. The conjecture is equivalent to the statement that every algebra norm on

*C*(

*X*) is equivalent to the usual uniform norm. (Kaplansky himself had earlier shown that every

*complete*algebra norm on

*C*(

*X*) is equivalent to the uniform norm.)

*if one furthermore assumes*the validity of the continuum hypothesis, there exist compact Hausdorff spaces

*X*and discontinuous homomorphisms from

*C*(

*X*) to some Banach algebra, giving counterexamples to the conjecture.

## Kaplansky conjecture on Quadratic Forms[edit]

^{[1]}

^{[2]}

*m*.

^{[1]}. In 1999 Oleg Izhboldin built a field with u-invariant

*m*=9 that was the first example of an odd u-invariant.

^{[3]}In 2006 Alexander Vishik demonstrated fields with u-invariant $m={2}^{k}+1$ for any integer

*k*starting from 3.

^{[4]}

## References[edit]

- ^
^{a}^{b}Merkur'ev, A. S. (1991). 'Kaplansky conjecture in the theory of quadratic forms'.*J Math Sci*.**57**: 3489. doi:10.1007/BF01100118. **^**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,
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