Abstract
Let n be a fixed positive integer, R be a prime ring, D and G two derivations of R and L a noncentral Lie ideal of R. Suppose that there exists 0 ≠ a ∈ R such that a(D(u)u n−u n G(u)) = 0 for all u ∈ L, where n ≥ 1 is a fixed integer. Then one of the following holds:
-
1.
D = G = 0, unless R satisfies s 4;
-
2.
char (R) ≠ 2, R satisfies s 4, n is even and D = G;
-
3.
char (R) ≠ 2, R satisfies s 4, n is odd and D and G are two inner derivations induced by b, c respectively such that b + c ∈ C;
-
4.
char (R) = 2 and R satisfies s 4.
We also investigate the case when R is a semiprime ring.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
N. Argaç and V. De Filippis, Co-centralizing derivations and nilpotent values on Lie ideals, Indian J. Pure Appl. Math., 41(3) (2010), 475–483.
K. I. Beidar, Rings of quotients of semiprime rings, Vestnik Moskov. Univ. Ser. I Mat. Mekh. (Engl. Transl: Moscow Univ. Math. Bull.), 33 (1978), 36–42.
K. I. Beidar, W. S. Martindale III and A. V. Mikhalev, Rings with generalized identities, Pure and Applied Math., 196, Marcel Dekker, New York, 1996.
J. Bergen, I. N. Herstein and J. W. Kerr, Lie ideals and derivations of prime rings, J. Algebra, 71 (1981), 259–267.
M. Brešar, Centralizing mappings and derivations in prime rings, J. Algebra, 156 (1993), 385–394.
L. Carini and V. De Filippis, Identities with generalized derivations on prime rings and Banach algebras, Algebra Colloquium, 19(spec. 1) (2012), 971–986.
I. S. Chang, K. W. Jun and Y. S. Jung, On derivations in Banach algebras, Bull. Korean Math. Soc., 39 (4) (2002), 635–643.
C. L. Chuang, Hypercentral derivations, J. Algebra, 166(1) (1994), 34–71.
C. L. Chuang, GPIs having coefficients in Utumi quotient rings, Proc. Amer. Math. Soc., 103(3) (1988), 723–728.
B. Dhara, Annihilator condition on power values of derivations, Indian J. Pure Appl. Math., 42(5) (2011), 357–369.
B. Dhara, Power values of derivations with annihilator conditions on Lie ideals in prime rings, Comm. Algebra, 37(6) (2009), 2159–2167.
T. S. Erickson, W. S. Martindale III and J. M. Osborn, Prime nonassociative algebras, Pacific J. Math., 60 (1975), 49–63.
I. N. Herstein, Center-like elements in prime rings, J. Algebra, 60 (1979), 567–574.
I. N. Herstein, Rings with involution, Univ. of Chicago Press, Chicago, 1976.
I. N. Herstein, Topics in ring theory, Univ. of Chicago Press, Chicago, IL, 1969.
B. E. Johnson and A. M. Sinclair, Continuity of derivations and a problem of Kaplansky, Amer. J. Math., 90 (1968), 1067–1073.
V. K. Kharchenko, Differential identity of prime rings, Algebra and Logic., 17 (1978), 155–168.
C. Lanski and S. Montgomery, Lie structure of prime rings of characteristic 2, Pacific J. Math., 42 (1) (1972), 117–136.
T. K. Lee and Y. Zhou, An identity with generalized derivations, Journal of Algebra and its Applications, 8(3) (2009), 307–317.
T. K. Lee, Semiprime rings with differential identities, Bull. Inst. Math. Acad. Sinica, 20(1) (1992), 27–38.
P. H. Lee and T. L. Wong, Derivations cocentralizing Lie ideals, Bull. Inst. Math. Acad. Sinica, 23 (1995), 1–5.
W. S. Martindale III, Prime rings satisfying a generalized polynomial identity, J. Algebra, 12 (1969), 576–584.
E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc., 8 (1957), 1093–1100.
A. M. Sinclair, Jordan homomorphisms and derivations on semisimple Banch algebras, Proc. Amer. Math. Soc., 24 (1970), 209–214.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work is supported by a grant from National Board for Higher Mathematics (NBHM), India. Grant No. is NBHM/R.P. 26/ 2012/Fresh/1745 dated 15.11.12 and second author is supported by The Scientific and Technological Research Council of Turkey, TUBITAK, No. 110T586
Rights and permissions
About this article
Cite this article
Dhara, B., Argaç, N. & Pradhan, K.G. Annihilator condition of a pair of derivations in prime and semiprime rings. Indian J Pure Appl Math 47, 111–124 (2016). https://doi.org/10.1007/s13226-015-0166-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13226-015-0166-z