GENERALIZED DERIVATIONS IN PRIME RINGS AND NONCOMMUTATIVE BANACH ALGEBRAS

Title & Authors
GENERALIZED DERIVATIONS IN PRIME RINGS AND NONCOMMUTATIVE BANACH ALGEBRAS
De Filippis, Vincenzo;

Abstract
Let R be a prime ring of characteristic different from 2, C the extended centroid of R, and $\small{\delta}$ a generalized derivations of R. If [[$\small{\delta(x)}$, x], $\small{\delta(x)}$] = 0 for all $\small{x\;{\in}\;R}$ then either R is commutative or $\small{\delta(x)\;=\;ax}$ for all $\small{x\;{\in}\;R}$ and some $\small{a\;{\in}\;C}$. We also obtain some related result in case R is a Banach algebra and $\small{\delta}$ is either continuous or spectrally bounded.
Keywords
prime ring;derivations;differential identities;Banach algebras;
Language
English
Cited by
1.
GENERALIZED DERIVATIONS WITH ANNIHILATOR CONDITIONS IN PRIME RINGS,;

대한수학회보, 2011. vol.48. 5, pp.917-922
1.
Generalized derivations with power values in rings and Banach algebras, Journal of the Egyptian Mathematical Society, 2013, 21, 2, 75
2.
On prime and semiprime rings with generalized derivations and non-commutative Banach algebras, Proceedings - Mathematical Sciences, 2016, 126, 3, 389
3.
Engel type identities with generalized derivations in prime rings, Asian-European Journal of Mathematics, 2017, 1850055
4.
Generalized Jordan Derivations on Semiprime Rings and Its Applications in Range Inclusion Problems, Mediterranean Journal of Mathematics, 2011, 8, 3, 271
5.
Generalized Derivations of Rings and Banach Algebras, Communications in Algebra, 2013, 41, 3, 1188
6.
On Lie Ideals with Generalized Derivations and Non-commutative Banach Algebras, Bulletin of the Malaysian Mathematical Sciences Society, 2017, 40, 2, 747
7.
Banach algebra with generalized derivations, Asian-European Journal of Mathematics, 2016, 1750069
8.
GENERALIZED DERIVATIONS WITH ANNIHILATOR CONDITIONS IN PRIME RINGS, Bulletin of the Korean Mathematical Society, 2011, 48, 5, 917
9.
On commutativity of rings with generalized derivations, Journal of the Egyptian Mathematical Society, 2016, 24, 2, 151
10.
A result on generalized derivations on right ideals of prime rings, Ukrainian Mathematical Journal, 2012, 64, 2, 186
References
1.
K. I. Beidar, Rings with generalized identities, Moscow Univ. Math. Bull. 33 (1978), no. 4, 53-58

2.
K. I. Beidar, W. S. Martindale III, and V. Mikhalev, Rings with Generalized Identities, Monographs and Textbooks in Pure and Applied Mathematics, 196. Marcel Dekker, Inc., New York, 1996

3.
M. Bresar and M. Mathieu, Derivations mapping into the radical. III, J. Funct. Anal. 133 (1995), no. 1, 21-29

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

5.
T. S. Erickson, W. S. Martindale III, and J. M. Osborn, Prime nonassociative algebras, Pacific J. Math. 60 (1975), no. 1, 49-63

6.
C. Faith and Y. Utumi, On a new proof of Litoff's theorem, Acta Math. Acad. Sci. Hungar 14 (1963), 369-371

7.
B. Hvala, Generalized derivations in rings, Comm. Algebra 26 (1998), no. 4, 1147-1166

8.
N. Jacobson, PI-algebras, Lecture Notes in Mathematics, Vol. 441. Springer-Verlag, Berlin-New York, 1975

9.
N. Jacobson, Structure of Rings, American Mathematical Society Colloquium Publications, Vol. 37. Revised edition American Mathematical Society, Providence, R.I. 1964

10.
B. E. Johnson and A. M. Sinclair, Continuity of derivations and a problem of Kaplansky, Amer. J. Math. 90 (1968), 1067-1073

11.
V. K. Kharchenko, Differential identities of prime rings, Algebra and Logic 17 (1978), no. 2, 155-168

12.
B.-D. Kim, Derivations of semiprime rings and noncommutative Banach algebras, Commun. Korean Math. Soc. 17 (2002), no. 4, 607-618

13.
T. K. Lee, Generalized derivations of left faithful rings, Comm. Algebra 27 (1999), no. 8, 4057-4073

14.
T. K. Lee, Semiprime rings with differential identities, Bull. Inst. Math. Acad. Sinica 20 (1992), no. 1, 27-38

15.
W. S. Martindale III, Prime rings satisfying a generalized polynomial identity, J. Algebra 12 (1969), 576-584

16.
M. Mathieu and G. J. Murphy, Derivations mapping into the radical, Arch. Math. (Basel) 57 (1991), no. 5, 469-474

17.
M. Mathieu and V. Runde, Derivations mapping into the radical. II, Bull. London Math. Soc. 24 (1992), no. 5, 485-487

18.
K.-H. Park, On derivations in noncommutative semiprime rings and Banach algebras, Bull. Korean Math. Soc. 42 (2005), no. 4, 671-678

19.
E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093-1100

20.
A. M. Sinclair, Continuous derivations on Banach algebras, Proc. Amer. Math. Soc. 20 (1969), 166-170

21.
I. M. Singer and J. Werner, Derivations on commutative normed algebras, Math. Ann. 129 (1955), 260-264

22.
M. P. Thomas, The image of a derivation is contained in the radical, Ann. of Math. (2) 128 (1988), no. 3, 435-460

23.
J. Vukman, A result concerning derivations in noncommutative Banach algebras, Glas. Mat. Ser. III 26(46) (1991), no. 1-2, 83-88