• Journal of the Korean Mathematical Society
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• v.47 no.3
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• pp.483-494
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• 2010

Let R be a non-commutative prime ring and I a non-zero left ideal of R. Let U be the left Utumi quotient ring of R and C be the center of U and k, m, n, r fixed positive integers. If there exists a generalized derivation g of R such that $[g(x^m)x^n,\;x^r]_k\;=\;0$ for all x $\in$ I, then there exists a $\in$ U such that g(x) = xa for all x $\in$ R except when $R\;{\cong}\;=M_2$(GF(2)) and I[I, I] = 0.