Identities related to generalized derivations in prime ∗-rings
Abstract Let {\mathcal{R}} be a prime ring with center {Z(\mathcal{R})} and {*} an involution of {\mathcal{R}} . Suppose that {\mathcal{R}} admits generalized derivations F, G and H associated with a nonzero derivation f, g and h of {\mathcal{R}} , respectively. In the present paper, we investigate the commutativity of a prime ring {\mathcal{R}} satisfying any of the following identities: (i) [F(x),F(x^{*})]=\nobreak 0 , (ii) [F(x),F(x^{*})]=\pm[x,x^{*}] , (iii) F(x)\circ\nobreak F(x^{*})=0 , (iv) F(x)\circ\nobreak F(x^{*})=\pm(x\circ\nobreak x^{*}) , (v) [F(x),x^{*}]\pm[x,G(x^{*})]=0 , (vi) F(xx^{*})\in Z(\mathcal{R}) , (vii) F(x)G(x^{*})\pm H(x)x^{*}\in Z(\mathcal{R}) , (viii) F([x,x^{*}])\pm[x,x^{*}]\in Z(\mathcal{R}) , (ix) F(x\circ\nobreak x^{*})\pm x\circ x^{*}\in Z(\mathcal{R}) , (x) [F(x),x^{*}]\pm[x,G(x^{*})]\in Z(\mathcal{R}) , (xi) F(x)\circ\nobreak x^{*}\pm x\circ\nobreak G(x^{*})\in Z(\mathcal{R}) for all {x\in\mathcal{R}} . Finally, the restrictions imposed on the hypotheses have been justified by an example.