Gurnınder Sıngh SANDHU, Didem KARALARLIOĞLU CAMCI

Some results on prime rings with multiplicative derivations

Let $R$ be a prime ring with center $Z R $ and an automorphism $\alpha.$ A mapping $\delta:R\to R$ is called multiplicative skew derivation if $\delta xy =\delta x y+ \alpha x \delta y $ for all $x,y\in R$ and a mapping $F:R\to R$ is said to be multiplicative generalized -skew derivation if there exists a unique multiplicative skew derivation $\delta$ such that $F xy =F x y+\alpha x \delta y $ for all $x,y\in R.$ In this paper, our intent is to examine the commutativity of $R$ involving multiplicative generalized -skew derivations that satisfy the following conditions: i $F x^{2} +x\delta x =\delta x^{2} +xF x $, ii $F x\circ y =\delta x\circ y \pm x\circ y$, iii $F [x,y] =\delta [x,y] \pm [x,y]$, iv $F x^{2} =\delta x^{2} $, v $F [x,y] =\pm x^{k}[x,\delta y ]x^{m}$, vi $F x\circ y =\pm x^{k} x\circ\delta y x^{m}$, vii $F [x,y] =\pm x^{k}[\delta x ,y]x^{m}$, viii $F x\circ y =\pm x \delta x \circ y x^{m}$ for all $x,y\in R.$

Keywords:

Prime ring, multiplicative generalized derivation multiplicative generalized -skew derivation, multiplicative left centralizer.,

___

[1] Bresˇar M. On the distance of the composition of two derivations to the generalized derivations. Glasgow Mathematical Journal 1991; 33: 89-93. doi: 10.1017/S0017089500008077
[2] Bresˇar M. Semiderivations of prime rings. Proceedings of American Mathematical Society 1990; 108 (4): 859-860. doi: 10.1090/S0002-9939-1990-1007488-X
[3] Camci DK, Aydin N. On multiplicative (generalized)-derivations in semiprime rings. Communications Faculty of Science University of Ankara Series A1 Maththematics and Statistics 2017; 66 (1): 153-164. doi: 10.1501/Commua1_0000000784
[4] Daif MN. When is a multiplicative derivation additive? International Journal of Mathematics and Mathematical Sciences 1991; 14 (3): 615-618. doi: 10.1155/S0161171291000844
[5] Daif MN, Tammam El-Sayiad MS. Multiplicative generalized derivations which are additive. East-West Journal of Mathematics 1997; 9 (1): 31–37.
[6] Daif MN, Bell HE. Remarks on derivations on semiprime rings. International Journal of Mathematics and Mathematical Sciences 1992; 15 (1): 205–206. doi: 10.1155/S0161171292000255
[7] Dhara B, Ali S. On multiplicative (generalized)-derivations in prime and semiprime rings. Aequationes Mathematicae 2013; 86 (12): 65-79. doi: 10.1007/s00010-013-0205-y
[8] Eremita D, Ilisˇević D. On additivity of centralisers. Bulletin of the Australian Mathematical Society 2006; 74 (2): 177-184. doi: 10.1017/S0004972700035620
[9] Goldmann H, S˘emrl P. Multiplicative derivations on C(X). Monatshefte fu¨r Mathematik 1996; 121: 189–197. doi: 10.1007/BF01298949
[10] Herstein IN. Rings with involutions. Chicago, USA: University of Chicago Press, 1969.
[11] Koç E, Gölbaşi O. Some results on ideals of semiprime rings with multiplicative generalized derivations. Communications in Algebra 2018; 46 (11): 4905–4913. doi: 10.1080/00927872.2018.1459644
[12] Lee TK, Shiue WK. A result on derivations with Engel condition in prime rings. Southeast Asian Bulletin of Mathematics 1999; 23: 437-446.
[13] Luh J. A note on commuting automorphisms of rings. American Mathematical Monthly 1970; 77: 61-62. doi: 10.1080/00029890.1970.11992420
[14] Quadri MA, Khan MS, Rehman NU. Generalized derivations and commutativity of prime rings. Indian Journal of Pure and Applied Mathematics 2003; 34 (9): 1393-1396.
[15] Rehman NU, Khan MS. A note on multiplicative (generalized)-skew derivation on semiprime rings. Journal of Taibah University of Sciences 2018; 12 (4): 450-454. doi: 10.1080/16583655.2018.1490049
[16] Sandhu GS, Kumar D. Derivable mappings and commutativity of associative rings. Italian Journal of Pure and Applied Mathematics 2018; 40: 376-393.
[17] Shang Y. A note on the commutativity od prime near-rings. Algebra Colloquium 2015; 22 (3): 361-366. doi: 10.1142/S1005386715000310