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}}
.