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Let $R$ be a commutative ring of characteristic $n\geq 0$ and $G$ be a group. It is known that the group ring $RG$ is bounded Lie Engel if and only if either $G$ is nilpotent and $G$ has a $p$-abelian normal subgroup of finite $p$-power index (if $n$ is a power of a prime $p$) or $G$ is abelian. In this paper we try to generalize this result; if $x$ and $y$ are elements of $RG$, let $[x,y]=xy-yx$ and inductively, $[x,\,_k\, y]=[[x,\,_{k-1}\, y],y]$. Let $m$ and $n$ be two natural numbers. Among other results, we show that if $RG$ satisfies $[x^m,\,_n\,y]=0$, then either $G'$ is a $p$-group (if $n$ is a power of a prime $p$) or $G$ is abelian. If $G$ is locally finite, then we show that $RG$ satisfies $[x^{m(x,y)},\,_{n(x,y)}\,y]=0$ if and only if either $G$ is locally nilpotent and $G'$ is a $p$-group (if $n$ is a power of a prime $p$) or $G$ is abelian.
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