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Let R be a prime ring of characteristic not equal to 2, U the Utumi quotient ring of R, $$C=Z(U)$$ the extended centroid of R, and $$f(x_1,\ldots ,x_n)$$ be a multilinear polynomial over C, not central valued on R. Suppose that F and G are two nonzero generalized derivations of R such that $$F(f(x_1,\ldots ,x_n))G(f(x_1,\ldots ,x_n))+G(f(x_1,\ldots ,x_n))F(f(x_1,\ldots ,x_n))=0$$ for all $$x_1,\ldots ,x_n\in R$$ . Then $$f(x_1,\ldots ,x_n)^2$$ is central valued on R and one of the following holds: [ABSTRACT FROM AUTHOR]

Let R be a prime ring with its Utumi ring of quotients U, C = Z(U) extended centroid of R, F a nonzero generalized derivation of R, L a noncentral Lie ideal of R and k ≥ 2 a fixed integer. Suppose that there exists 0 ≠ a ∊ R such that a|F(un1), un2, . . . , unk ] = 0 for all u ∊ L, where n1, n2, . . . , nk ≥ 1 are fixed integers. Then either there exists λ ≥ ∊ C such that F(x) = ≥x for all x ∊ R, or R satisfies s4, the standard identity in four variables. [ABSTRACT FROM AUTHOR]