No Access

Generalized derivations on Lie ideals in prime rings

V. K. Yadav

Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India

E-mail Address: vishalny87@gmail.com

Corresponding author.

Search for more papers by this author

S. K. Tiwari

Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India

E-mail Address: shaileshiitd84@gmail.com

Search for more papers by this author

, and

R. K. Sharma

Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India

E-mail Address: rksharma@maths.iitd.ac.in

Search for more papers by this author

https://doi.org/10.1142/S1793557117500322Cited by:1 (Source: Crossref)

Abstract

Let $R$ $R$ be a $2$ $2$ -torsion free prime ring, and $U$ $U$ a square closed Lie ideal of $R .$ $R .$ Further let $G,$ $G,$ and $F$ $F$ be generalized derivations associated with derivations $g$ $g$ and $d$ $d$ , respectively on $R .$ $R .$ If one of the following conditions holds: (i) $G (u v) + d (u) F (v) + u v \in Z (R),$ $G (u v) + d (u) F (v) + u v \in Z (R),$ (ii) $G (u v) + d (u) F (v) + v u \in Z (R),$ $G (u v) + d (u) F (v) + v u \in Z (R),$ (iii) $G (u v) + d (u) F (v) + u v \in Z (R),$ $G (u v) + d (u) F (v) + u v \in Z (R),$ (iv) $G (v u) + F (u) F (v) + u v \in Z (R),$ $G (v u) + F (u) F (v) + u v \in Z (R),$ (v) $G (u v) + d (v) F (u) + u v \in Z (R),$ $G (u v) + d (v) F (u) + u v \in Z (R),$ for all $u, v \in U,$ $u, v \in U,$ then it is proved that either $g = d = 0$ $g = d = 0$ or $U \subseteq R .$ $U \subseteq R .$

Communicated by V. K. Yadav

Keywords:

AMSC: 16W25, 16N60, 16R50

References

1. S. Ali, B. Dhara and A. Fošner , Some commutativity theorems concerning additive mappings and derivations on semiprime rings, in Proc. 6th China–Japan–Korea Conf., eds. Kwak et al. (World Scientific, Singapore, 2011), pp. 133–141. Google Scholar
2. A. Ali, D. Kumar and P. Miyan , On generalized derivations and commutativity of prime and semiprime rings, Hacettepe J. Math. Stat. 40(3) (2011) 367–374. Web of Science, Google Scholar
3. M. Ashraf, A. Ali and S. Ali , Some commutativity theorem for prime rings with generalized derivations, Southeast Asian Bull. Math. 31 (2007) 415–421. Google Scholar
4. M. Ashraf and N. Rehman , On derivations and commutativity in prime rings, East–West J. Math. 3(1) (2001) 87–91. Google Scholar
5. M. J. Atteya , On generalized derivations of semiprime rings, Int. J. Algebra 4(12) (2010) 591–598. Google Scholar
6. R. Awtar , Lie structure in prime rings with derivations, Publ. Math. Debrecen 31 (1984) 209–215. Crossref, Web of Science, Google Scholar
7. J. Bergen, I. N. Herstein and J. W. Kerr , Lie ideals and derivations of prime rings, J. Algebra 71 (1981) 259–267. Crossref, Web of Science, Google Scholar
8. M. Brešar , On the distance of the composition of two derivations to the generalized derivations, Glasgow Math. J. 33 (1991) 89–93. Crossref, Web of Science, Google Scholar
9. L. Carini and V. De Fillippis , Commutators with power central values on a Lie ideal, Pacific J. Math. 193 (2000) 269–278. Crossref, Web of Science, Google Scholar
10. B. Dhara, S. Kar and S. Mondal , A result on generalized derivations on Lie ideals in prime rings, Beitr. Algebra Geom. 54 (2013) 677–682. Crossref, Google Scholar
11. O. Gölbaşi and E. Koç , Notes on commutativity of prime rings with generalized derivation, Commun. Fac. Sci. Univ. 58 (2009) 39–46. Google Scholar
12. C. Lanski , A Engel condition with derivation, Proc. Amer. Math. Soc. 118 (1993) 731–734. Crossref, Web of Science, Google Scholar
13. H. Shuliang , Generalized derivations of ∗-prime rings, Int. J. Algebra 2(18) (2008) 867–873. Google Scholar
14. S. K. Tiwari, R. K. Sharma and B. Dhara , Multiplicative (generalized)-derivation in semiprime rings, Beitr. Algebra Geom. (2015) 1–15. Google Scholar
15. S. K. Tiwari, R. K. Sharma and B. Dhara , Identities related to generalized derivation on ideals in prime rings, Beitr. Algebra Geom. (2015), doi: 10.1007/s13366-015-0262-6. Google Scholar
16. Y. Wang and H. You , A note on commutators with power central values on Lie ideals, Acta Math. Sin. (Engl. Ser.) 22(6) (2006) 1715–1720. Crossref, Web of Science, Google Scholar