Trace Functions in the Ring of Fractions of Polycyclic Group Rings

Abstract

Let $KG$ be the group ring of a polycyclic-by-finite group $G$ over a field $K$ of characteristic zero, $R$ be the Goldie ring of fractions of $KG$, $S$ be an arbitrary subring of ${R_{n \times n}}$. We prove that the intersection of the commutator subring $[S,S]$ with the center $Z(S)$ is nilpotent. This implies the existence of a nontrivial trace function in ${R_{n \times n}}$.