Notes on Generalized Derivations on Lie Ideals in Prime Rings

Let R be a prime ring, H a generalized derivation of R and L a noncommutative Lie ideal of R. Suppose that usH(u)ut = 0 for all u ∈ L, where s ≥ 0, t ≥ 0 are fixed integers. Then H(x) = 0 for all x ∈ R unless char R = 2 and R satisfies S4, the standard identity in four variables. Let R be an associative ring with center Z(R). For x, y ∈ R, the commutator xy− yx will be denoted by [x, y]. An additive mapping d from R to R is called a derivation if d(xy) = d(x)y + xd(y) holds for all x, y ∈ R. A derivation d is inner if there exists a ∈ R such that d(x) = [a, x] holds for all x ∈ R. An additive subgroup L of R is said to be a Lie ideal of R if [u, r] ∈ L for all u ∈ L, r ∈ R. The Lie ideal L is said to be noncommutative if [L,L] 6= 0. Hvala [8] introduced the notion of generalized derivation in rings. An additive mapping H from R to R is called a generalized derivation if there exists a derivation d from R to R such that H(xy) = H(x)y +xd(y) holds for all x, y ∈ R. Thus the generalized derivation covers both the concepts of derivation and left multiplier mapping. The left multiplier mapping means an additive mapping F from R to R satisfying F (xy) = F (x)y for all x, y ∈ R. Throughout this paper R will always present a prime ring with center Z(R), extended centroid C and U its Utumi quotient ring. It is well known that if ρ is a right ideal of R such that u = 0 for all u ∈ ρ, where n is a fixed positive integer, then ρ = 0 [7, Lemma 1.1]. In [2], Chang and Lin consider the situation when d(u)u = 0 for all u ∈ ρ and ud(u) = 0 for all u ∈ ρ, where ρ is a nonzero right ideal of R. More precisely, they proved the following: Let R be a prime ring, ρ a nonzero right ideal of R, d a derivation of R and n a fixed positive integer. If d(u)u = 0 for all u ∈ ρ, then d(ρ)ρ = 0 and if ud(u) = 0 for all u ∈ ρ, then d = 0 unless R ∼= M2(F ), the 2×2 matrices over a field F of two elements. Received July 28, 2008. 2000 Mathematics Subject Classification. 16W25, 16N60, 16R50.