NOTES ON (σ, τ)-DERIVATIONS OF LIE IDEALS IN PRIME RINGS

Abstract. Let R be a prime ring with center Z and characteristic differ-ent from two, U a nonzero Lie ideal of R such that u2 ∈ U for all u ∈ U and d be a nonzero (σ, τ)-derivation of R. We prove the following results: (i) If [d(u), u]σ,τ = 0 or [d(u), u]σ,τ ∈ Cσ,τ for all u ∈ U, then U ⊆ Z. (ii) If a ∈ R and [d(u), a]σ,τ = 0 for all u ∈ U, then U ⊆ Z or a ∈ Z. (iii) If d([u, v]) = ±[u, v]σ,τ for all u ∈ U, then U ⊆ Z. 1