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Let R be a prime ring, with extended centroid C, g a non-zero generalized derivation of R, L a non-central Lie ideal of R, k ≥ 1 a fixed integer. If [g(u), u] k = 0, for all u, then either g(x) = ax, with a ∈ C or R satisfies the standard identity s 4 . Moreover in the latter case eitherWe also prove a more generalized version by replacing L with the set [I, I], where I is a right ideal of R.

Let R be a ring which possesses a unit element, a Lie ideal U ⊄ Z, and a derivation d such that d(U) ≠ 0 and d(u) is 0 or invertible, for all u ∈ U. We prove that R must be either a division ring D or D2, the 2 X 2 matrices over a division ring unless d is not inner, R is not semiprime, and either 2R or 3R is 0. We also examine for which division rings D, D2 can possess such a derivation and study when this derivation must be inner.

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