Identities related to generalized derivations in prime ∗-rings

Abstract

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 * ) = 0 F(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.