Search

Let R be a semiprime ring with characteristic different from two and L a noncentral square-closed Lie ideal of R. Suppose that R admits a multiplicative generalized derivation (F,d) satisfying d(L) ⊆ L. In the present paper, we shall prove that d is commuting on L if one of the following conditions holds: (i) F([x,y]) = ∓ [x,d(y)]; (ii) F(x ∘ y) = ∓ (x ∘ d(y)); (iii) F([x,y]) = ∓ (F(y)x); (iv) F([x,y]) = ∓ (F(x)y); (v) F(x ∘ y) = ∓ (F(y)x); (vi) F(x ∘ y) = ∓ (F(x)y) for all x,y ∈ L.

Let M be a prime Γ-ring with center Z(M), I a nonzero ideal of M and F be a generalized derivation with associated nonzero derivation d. In the present paper, our purpose is to produce commutativity results for prime Γ-rings M admitting a generalized derivation F satisfying any one of the properties: (i) F(xαy)∓xαy ∈ Z(M), (ii) F(xαy)∓yαx ∈ Z(M), (iii) F(x)αF(y)∓xαy ∈ Z(M), (iv) F(x)αF(y)∓yαx ∈ Z(M), (v) F([x, y]α)=[F(x), y]α, (vi) [F(x), y]α∓[x, F(y)]α=0, (vii) F([x, y]α)∓[d(x), d(y)]α=0, for all x,y ∈ I and α ∈ Γ. Also, some examples are given to show that the primeness of the various results is not superfluous.