Commutativity Theorems on Prime Rings with Generalized Derivations

Suppose that \(\mathcal {R}\) is a prime ring, \(\mathcal {I}\) a nonzero ideal of \(\mathcal {R}\) , \(\mathcal {F}\) a generalized derivation of \(\mathcal {R}\) and n a fixed positive integer. If \( (\mathcal {F}(x_1)x_2+x_1\mathcal {F}(x_2)+\mathcal {F}(x_2)x_1+x_2\mathcal {F}(x_1))^n-(x_1x_2+x_2x_1)=0, \) for all \(x_1,x_2\in \mathcal {I}\) , then one of the following holds: If \(char(\mathcal {R})\ne 2\) and \( (\mathcal {F}(x_1)x_2+x_1\mathcal {F}(x_2)+\mathcal {F}(x_2)x_1+x_2\mathcal {F}(x_1))^n-(x_1x_2+x_2x_1)\in \mathcal {Z}(\mathcal {R}), \) for all \(x_1,x_2\in \mathcal {I}\) , then one of the following holds: We examine the aforementioned identities in semiprime rings and also obtain some range inclusion results on Banach algebras.