Let R be a semiprime ring with center Z(R). For x; y 2 R, we denote by [x; y] = xy - yx the commutator of x and y. If is a non-identity automorphism of R such that ⋯ [(xn0 ); xn1 ]; xn2 ; ⋯ ; xnk = 0 for all x 2 R, where n0; n1; n2; ⋯ ; nk are fixed positive integers, then there exists a map : R ! Z(R) such that (x) = x + (x) for all x 2 R. In particular, when R is a prime ring, R is commutative.
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