10.41 (1986)

(Well-known problem). Let $\Gamma$ be an almost polycyclic group with no non-trivial finite normal subgroups, and let $k$ be a field. The complete ring of quotients $Q(k\Gamma)$ is a matrix ring $M_n(D)$ over a skew field. Conjecture: $n$ is the least common multiple of the orders of the finite subgroups of $\Gamma$. An equivalent formulation (M. Lorenz) is as follows: $\rho(G_0(k\Gamma)) = \rho(G_0(k\Gamma)_{\mathcal{F}})$, where $G_0(k\Gamma)$ is the Grothendieck group of the category of finitely-generated $k\Gamma$-modules, $G_0(k\Gamma)_{\mathcal{F}}$ is the subgroup generated by classes of modules induced from finite subgroups of $\Gamma$, and $\rho$ is the Goldie rank. There is a stronger conjecture: $G_0(k\Gamma) = G_0(k\Gamma)_{\mathcal{F}}$.