ABSTRACT. We extend a result of Herstein co,cerning a derivation d on a prime ring R satisfying [d(x),d(y)] 0 for all x,y R, to the case of semiprime rings. An extension of this result is proved for a two-sided ideal but is shown to be not true for a one-sided ideal. Some of our recent results dealing with U*-and U**-derivations on a prime ring are extended to semiprime rings. Finally, we obtain a result on semiprime rings for which d(xy) d(yx) for all x,y in some ideal U.
Let R be an associative prime ring, U a Lie ideal such that u 2 ∈U for all u∈U and F: R→R be a generalized derivation. In this paper, we show that U ⊆ Z(R) if any one of the following conditions holds: (i) F(uv) -uv∈Z(R),(ii) F(uv) -vu∈Z(R), (iii) uF(v) + uv∈Z(R), and (iv) [F(u), v] + uv∈Z(R) for all u, v∈U. If we choose the underlying subset of R as an ideal instead of a Lie ideal of R, then we prove the commutativity of prime ring.
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.
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