Let R be an associative ring with center Z (R) and d a nonzero derivation of R. The main object in this paper is to study the situation [[d(x r )x n , x r ] s , [y , d(y )] t ] m ∈ Z(R) for all x, y in some appropriate subset of R, where n ≥ 0, s ≥ 0, t ≥ 0, m ≥ 1, r ≥ 1 are fixed integers and R is a prime or semiprime ring.
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