Trace functions in the ring of fractions of polycyclic group rings

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