Geodesic
Generalized reverse derivations and commutativity of prime rings
Communications in Mathematics, Tome 27 (2019) no. 1, pp. 43-50.

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Let $R$ be a prime ring with center $Z(R)$ and $I$ a nonzero right ideal of $R$. Suppose that $R$ admits a generalized reverse derivation $(F,d)$ such that $d(Z(R))\neq 0$. In the present paper, we shall prove that if one of the following conditions holds: (i) $F(xy)\pm xy\in Z(R)$, (ii) $F([x,y])\pm [F(x),y]\in Z(R)$, (iii) $F([x,y])\pm [F(x),F(y)]\in Z(R)$, (iv) $F(x\circ y)\pm F(x)\circ F(y)\in Z(R)$, (v) $[F(x),y]\pm [x,F(y)]\in Z(R)$, (vi) $F(x)\circ y\pm x\circ F(y)\in Z(R)$ for all $x,y \in I$, then $R$ is commutative.
Classification : 16A70, 16N60, 16W25
Keywords: Prime rings; reverse derivations; generalized reverse derivations. 
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     author = {Huang, Shuliang},
     title = {Generalized reverse derivations and commutativity of prime rings},
     journal = {Communications in Mathematics},
     pages = {43--50},
     publisher = {mathdoc},
     volume = {27},
     number = {1},
     year = {2019},
     mrnumber = {3977476},
     zbl = {07368958},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/COMIM_2019__27_1_a3/}
}
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Huang, Shuliang. Generalized reverse derivations and commutativity of prime rings. Communications in Mathematics, Tome 27 (2019) no. 1, pp. 43-50. http://geodesic.mathdoc.fr/item/COMIM_2019__27_1_a3/