Mathematics – Rings and Algebras
Scientific paper
2002-02-04
in: High-dimensional manifold topology. Proceedings of the school held in Trieste, May 21--June 8, 2001. Edited by F. T. Farre
Mathematics
Rings and Algebras
LaTex2e, 7 pages. Added a new proof of the main result (due to Warren Dicks) which is shorter, easier and more elementary, and
Scientific paper
Let G be the wreath product of Z and Z/2, the so called lamplighter group and k a commutative ring. We show that kG does not have a classical ring of quotients (i.e. does not satisfy the Ore condition). This answers a Kourovka notebook problem. Assume that kG is contained in a ring R in which the element 1-x is invertible, with x a generator of Z considered as subset of G. Then R is not flat over kG. If k is the field of complex numbers, this applies in particular to the algebra UG of unbounded operators affiliated to the group von Neumann algebra of G. We present two proofs of these results. The second one is due to Warren Dicks, who, having seen our argument, found a much simpler and more elementary proof, which at the same time yielded a more general result than we had originally proved. Nevertheless, we present both proofs here, in the hope that the original arguments might be of use in some other context not yet known to us.
Linnell Peter
Lueck Wolfgang
Schick Thomas
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