Let
A
A
be an additive subgroup of a group ring
R
R
over a field
K
K
. Denote by
[
A
,
R
]
[A,R]
the additive subgroup generated by the Lie products
[
a
,
r
]
=
a
r
−
r
a
,
a
∈
A
,
r
∈
R
[a,r] = ar - ra,a \in A,r \in R
. Inductively, let
[
A
,
R
n
]
=
[
[
A
,
R
n
−
1
]
,
R
]
[A,{R_n}] = [[A,{R_{n - 1}}],R]
. We prove that
[
A
,
R
n
]
=
0
[A,{R_n}] = 0
for some
n
⇒
[
A
,
R
]
R
n \Rightarrow [A,R]R
is a nilpotent ideal.