Generalized Lie nilpotent group rings

Let $KG$ be the group ring of a group $G$ over a ring $K$ with identity. The ring $KG$ is said to be Lie $T$-nilpotent if for every sequence $x_1,x_2,\dots,x_n,\dots$ of elements of $KG$ there is an index $m$ such that the Lie commutator $(\dots((x_1,x_2),x_3)\dots,x_m)=0$. It is proved that $KG$ is a Lie $T$-nilpotent ring if and only if $K$ is Lie $T$-nilpotent and one of the following conditions is satisfied: 1) $G$ is an Abelian group, or 2) $K$ is a ring of characteristic $p^m$ ($p$ prime), $G$ is a nilpotent group and its commutator subgroup is a finite $p$-group.
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