Geodesic
Generalized reverse derivations and commutativity of prime rings
Communications in Mathematics, Tome 27 (2019) no. 1, pp. 43-50
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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. 
@article{COMIM_2019_27_1_a3,
     author = {Huang, Shuliang},
     title = {Generalized reverse derivations and commutativity of prime rings},
     journal = {Communications in Mathematics},
     pages = {43--50},
     year = {2019},
     volume = {27},
     number = {1},
     mrnumber = {3977476},
     zbl = {07368958},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/COMIM_2019_27_1_a3/}
}
TY  - JOUR
AU  - Huang, Shuliang
TI  - Generalized reverse derivations and commutativity of prime rings
JO  - Communications in Mathematics
PY  - 2019
SP  - 43
EP  - 50
VL  - 27
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/COMIM_2019_27_1_a3/
LA  - en
ID  - COMIM_2019_27_1_a3
ER  - 
%0 Journal Article
%A Huang, Shuliang
%T Generalized reverse derivations and commutativity of prime rings
%J Communications in Mathematics
%D 2019
%P 43-50
%V 27
%N 1
%U http://geodesic.mathdoc.fr/item/COMIM_2019_27_1_a3/
%G en
%F COMIM_2019_27_1_a3
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/

[1] Aboubakr, A., Gonzalez, S.: Generalized reverse derivations on semiprime rings. Siberian Math. J., 56, 2, 2015, 199-205, | DOI | MR

[2] Albas, E.: On generalized derivations satisfying certain identities. Ukrainian Math. J., 63, 5, 2001, 699-698, | MR

[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

[4] Ali, A., Shah, T.: Centralizing and commuting generalized derivations on prime rings. Matematiqki Vesnik, 60, 2008, 1-2, | MR

[5] Ashraf, M., Ali, A., Ali, S.: Some commutativity theorems for rings with generalized derivations. Southeast Asian Bull. Math., 31, 2007, 415-421, | MR

[6] Ashraf, M., Rehman, N.: Derivations and commutativity in prime rings. East-West J. Math., 3, 1, 2001, 87-91, | MR

[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

[8] Filippov, V.T.: $\delta $-derivations of prime Lie algebras. Siberian Math. J., 40, 1, 1999, 174-184, | DOI | MR

[9] Herstein, I.N.: Jordan derivations of prime rings. Proc. Amer. Math., Soc., 8, 1957, 1104-1110, | MR

[10] Hopkins, N.C.: Generalized derivations of nonassociative algebras. Nova J. Math. Game Theory Algebra, 5, 3, 1996, 215-224, | MR

[11] Huang, S.: Notes on commutativity of prime rings. Algebra and its Applications, 174, 2016, 75-80, | MR

[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 | MR | Zbl

[14] Posner, E.C.: Derivations in prime rings. Proc. Amer. Math. Soc., 8, 1957, 1093-1100, | MR

[15] Samman, M., Alyamani, N.: Derivations and reverse derivations in semiprime rings. Int. J. Forum, 39, 2, 2007, 1895-1902, | DOI | MR

[16] Reddy, C.J. Subba, Hemavathi, K.: Right reverse derivations on prime rings. Int. J. Res. Eng. Tec., 2, 3, 2014, 141-144,