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Throughout this note, \(R\) denotes an associative ring and \(C(R)\) be the center of \(R\). In this paper, it is
proved that a non-central Lie ideal \(L\) of a semiprime ring \(R\) contains a nonzero ideal of \(R\) and this result is
used to obtain several commutativity theorems of \(R\) involving multiplicative derivations. Moreover, some
results on one-sided ideals of \(R\) are given.