LAUSR

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