  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.
