Let
$R$
be a semiprime ring with center
$Z\left( R \right)$
. For
$x,\,y\,\in \,R$
, we denote by
$\left[ x,\,y \right]\,=\,xy\,-\,yx$
the commutator of
$x$
and
$y$
. If
$\sigma $
is a non-identity automorphism of
$R$
such that
1
$$\left[ \left[ \cdot \cdot \cdot \,\left[ \left[ \sigma \left( {{x}^{n0}} \right),\,{{x}^{n1}} \right],\,{{x}^{n2}} \right],\cdot \cdot \cdot \right],\,{{x}^{nk}} \right]\,=\,0$$
for all
$x\,\in \,R$
, where
${{n}_{0}},\,{{n}_{1}},\,{{n}_{2}},\,...,\,{{n}_{k}}$
are fixed positive integers, then there exists a map
$\mu \,:\,R\,\to \,Z\left( R \right)$
such that
$\sigma \left( x \right)\,=\,x\,+\,\mu \left( x \right)$
for all
$x\,\in \,R$
. In particular, when
$R$
is a prime ring,
$R$
is commutative.