Generalized reverse derivations and commutativity of prime rings

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Abstract

Let R be a prime ring with center Z(R) and I a nonzero right ideal of R. Suppose that R admits a generalized reverse derivation (F, d) such that d(Z(R)) ≠ 0. In the present paper, we shall prove that if one of the following conditions holds:

(i) F (xy) ± xyZ(R)

(ii) F ([x, y]) ± [F (x), y] ∈ Z(R)

(iii) F ([x, y]) ± [F (x), F (y)] ∈ Z(R)

(iv) F (x ο y) ± F (x) ο F (y) ∈ Z(R)

(v) [F (x), y] ± [x, F (y)] ∈ Z(R)

(vi) F (x) ο y ± x ο F (y) ∈ Z(R)

for all x, yI, then R is commutative.

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