Let R be a prime ring with a nonzero derivation d and let f (X-1,..., X-t) be a multilinear polynomial over C, the extended centroid of R. Suppose that b [d (f (x(1),..., x(t))), f(x(1),..., x(t))](n) = 0 for all x(i) is an element of R, where 0 not equal b is an element of R and n is a fixed positive integer. Then f(X-1,..., X-t) is centrally valued on R unless char R = 2 and dim(C)RC = 4. We prove a more generalized version by replacing R with a left ideal.
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