On the trace of an idempotent in a group ring

Abstract:

Let KG be the group ring of a polycyclic by finite group G over a field K of characteristic zero. It is proved that if $e = \sum e(g)g$ is a nontrivial idempotent in KG then its trace $e(1)$ is a rational number $r/s,(r,s) = 1$, with the property that for every prime divisor p of s, G has an element of order p. This result is used to prove that if R is a commutative ring of characteristic zero, without nontrivial idempotents and G is a polycyclic by finite group such that no group order $\ne 1$ is invertible in R, then RG has no nontrivial idempotents.