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###### Generalized reverse derivations and commutativity of prime rings

## 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*) ± *xy* ∈ *Z*(*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, y* ∈ *I*, then *R* is commutative.