GENERALIZED DERIVATIONS ON SEMIPRIME RINGS

Title & Authors
GENERALIZED DERIVATIONS ON SEMIPRIME RINGS
De Filippis, Vincenzo; Huang, Shuliang;

Abstract
Let R be a prime ring, I a nonzero ideal of R and n a fixed positive integer. If R admits a generalized derivation F associated with a derivation d such that c for all x, $\small{y{\in}I}$. Then either R is commutative or n = 1, d = 0 and F is the identity map on R. Moreover in case R is a semiprime ring and $\small{(F([x,\;y]))^n=[x,\;y]}$ for all x, $\small{y{\in}R}$, then either R is commutative or n = 1, $\small{d(R){\subseteq}Z(R)}$, R contains a non-zero central ideal and for all $\small{x{\in}R}$.
Keywords
prime and semiprime rings;differential identities;generalized derivations;
Language
English
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3.
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References
1.
N. Argac and H. G. Inceboz, Derivations of prime and semiprime rings, J. Korean Math. Soc. 46 (2009), no. 5, 997-1005.

2.
M. Ashraf and N. Rehman, On commutativity of rings with derivations, Results Math. 42 (2002), no. 1-2, 3-8.

3.
K. I. Beidar, W. S. Martindale, and V. Mikhalev, Rings with Generalized Identities, Monographs and Textbooks in Pure and Applied Mathematics, 196. Marcel Dekker, Inc., New York, 1996.

4.
C. L. Chuang, GPI's having coefficients in Utumi quotient rings, Proc. Amer. Math. Soc. 103 (1988), no. 3, 723-728.

5.
M. N. Daif and H. E. Bell, Remarks on derivations on semiprime rings, Internt. J. Math. Math. Sci. 15 (1992), 205-206.

6.
J. S. Erickson, W. S. Martindale III, and J. M. Osborn, Prime nonassociative algebras, Pacific J. Math. 60 (1975), no. 1, 49-63.

7.
A. Giambruno and I. N. Herstein, Derivations with nilpotent values, Rend. Circ. Mat. Palermo (2) 30 (1981), no. 2, 199-206.

8.
B. Hvala, Generalized derivations in rings, Comm. Algebra 26 (1998), no. 4, 1147-1166.

9.
V. K. Kharchenko, Differential identities of prime rings, Algebra and Logic 17 (1978), 155-168.

10.
C. Lanski, An Engel condition with derivation, Proc. Amer. Math. Soc. 118 (1993), no. 3, 731-734.

11.
T. K. Lee, Generalized derivations of left faithful rings, Comm. Algebra 27 (1999), no. 8, 4057-4073.

12.
T. K. Lee, Semiprime rings with differential identities, Bull. Inst. Math. Acad. Sinica 20 (1992), no. 1, 27-38.

13.
W. S. Martindale III, Prime rings satisfying a generalized polynomial identity, J. Algebra 12 (1969), 176-584.

14.
M. A. Quadri, M. S. Khan, and N. Rehman, Generalized derivations and commutativity of prime rings, Indian J. Pure Appl. Math. 34 (2003), no. 9, 1393-1396.

15.
B. Zalar, On centralizers of semiprime rings, Comment. Math. Univ. Carolin. 32 (1991), no. 4, 609-614.