Let R be a prime ring with extended centroid C, λ a nonzero left ideal of R and f(X1, . . ., Xt) a nonzero multilinear polynomial over C. Suppose that d and δ are derivations of R such that d(f(x1,...,xt))f(x1,...,xt)-f(x1,...,xt)δ(f(x1,...,xt))ε C for all x1,...,xt ε λ. Then either d = 0 and λδ(λ) = 0 or λC = RCe for some idempotent e in the socle of RC and one of the following holds:(1) f(X1, . . ., Xt) is central-valued on eRCe;(2) λ(d + δ)(λ) = 0 and f (X1, . . ., Xt)2 is central-valued on eRCe;(3) char R = 2 and eRCe satisfies st4(X1, X2, X3, X4), the standard polynomial identity of depsiee 4.
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