# Article

Full entry | PDF   (0.2 MB)
Keywords:
Prime rings; reverse derivations; generalized reverse derivations.
Summary:
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.
References:
[1] Aboubakr, A., Gonzalez, S.: Generalized reverse derivations on semiprime rings. Siberian Math. J., 56, 2, 2015, 199-205, DOI 10.1134/S0037446615020019 | MR 3381237
[2] Albas, E.: On generalized derivations satisfying certain identities. Ukrainian Math. J., 63, 5, 2001, 699-698, MR 3093034
[3] Ali, A., Kumar, D., Miyan, P.: On generalized derivations and commutativity of prime and semiprime rings. Hacettepe J. Math. Statistics, 40, 3, 2011, 367-374, MR 2857629
[4] Ali, A., Shah, T.: Centralizing and commuting generalized derivations on prime rings. Matematiqki Vesnik, 60, 2008, 1-2, MR 2403266
[5] Ashraf, M., Ali, A., Ali, S.: Some commutativity theorems for rings with generalized derivations. Southeast Asian Bull. Math., 31, 2007, 415-421, MR 2327138
[6] Ashraf, M., Rehman, N.: Derivations and commutativity in prime rings. East-West J. Math., 3, 1, 2001, 87-91, MR 1866647
[7] Ashraf, M., Rehman, N., Mozumder, M.R.: On semiprime rings with generalized derivations. Bol. Soc. Paran. de Mat., 28, 2, 2010, 25-32, MR 2727428
[8] Filippov, V.T.: $\delta$-derivations of prime Lie algebras. Siberian Math. J., 40, 1, 1999, 174-184, DOI 10.1007/BF02674305 | MR 1686989
[9] Herstein, I.N.: Jordan derivations of prime rings. Proc. Amer. Math., Soc., 8, 1957, 1104-1110, MR 0095864
[10] Hopkins, N.C.: Generalized derivations of nonassociative algebras. Nova J. Math. Game Theory Algebra, 5, 3, 1996, 215-224, MR 1455818
[11] Huang, S.: Notes on commutativity of prime rings. Algebra and its Applications, 174, 2016, 75-80, MR 3613783
[12] Ibraheem, A.M.: Right ideal and generalized reverse derivations on prime rings. Amer. J. Comp. Appl. Math., 6, 4, 2016, 162-164,
[13] Mayne, J.H.: Centralizing mappings of prime rings. Canad. Math. Bull., 27, 1984, 122-126, DOI 10.4153/CMB-1984-018-2 | MR 0725261 | Zbl 0537.16029
[14] Posner, E.C.: Derivations in prime rings. Proc. Amer. Math. Soc., 8, 1957, 1093-1100, MR 0095863
[15] Samman, M., Alyamani, N.: Derivations and reverse derivations in semiprime rings. Int. J. Forum, 39, 2, 2007, 1895-1902, DOI 10.12988/imf.2007.07168 | MR 2341167
[16] Reddy, C.J. Subba, Hemavathi, K.: Right reverse derivations on prime rings. Int. J. Res. Eng. Tec., 2, 3, 2014, 141-144,

Partner of