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.

中文翻译：

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与素*环上的广义导数有关的恒等式

令ℛ{\ mathcal {R}}为质心环，中心为Z⁢（ℛ）{Z（\ mathcal {R}）}，* {*}为volution {\ mathcal {R}}的对合。假设ℛ{\ mathcal {R}}分别接受与ℛ{\ mathcal {R}}的非零导数f，g和h关联的广义导数F，G和H。在本文中，我们研究满足以下任意一个身份的素环ring {\ mathcal {R}}的可交换性：（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}）对于所有x∈ℛ{x \ in \ mathcal {R}}。最后，通过一个例子证明了对假设施加的限制。