Generalized reverse derivations and commutativity of prime rings

Let be a prime ring with center ) and a nonzero right ideal of . Suppose that admits a generalized reverse derivation ( , ) such that )) ≠ 0. In the present paper, we shall prove that if one of the following conditions holds: (i) ( ) ± ∈ (ii) ([ , ]) ± [ ( ), ] ∈ (iii) ([ ]) [ ( ), ( )] ∈ (iv) ( ο ) ± ( ) ο ( ) ∈ (v) [ ( ), ] ± [ , ( )] ∈ (vi) ( ) ο ο ( ) ∈ for all ∈ , then is commutative.