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Let be a noncommutative prime ring and
an automorphism of
,
. We denote by
the additive subgroup of
generated by the subset
and by
the subring of
generated by
. In this paper we prove that if
, then
contains
for some nonzero ideal
of
. Also, we prove the following theorem concerning
: Suppose that
. Then
contains a nonzero ideal of
if and only if either
or
.
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