Authors

Department of Mathematics, Islamic Azad University, Central Tehran Branch, P. O. Box 14168-94351, Tehran, Iran

Abstract

Let $R$ be a 2-torsion free semiprime ring with extended centroid $C$, $U$ the Utumi quotient ring of $R$ and $m,n>0$ are fixed integers. We show that if $R$ admits derivation $d$ such that $b[[d(x), x]_n,[y,d(y)]_m]=0$ for all $x,y\in R$ where $0\neq b\in R$, then there exists a central idempotent element $e$ of $U$ such that $eU$ is commutative ring and $d$ induce a zero derivation on $(1-e)U$. We also obtain some related result in case $R$ is a non-commutative Banach algebra and d continuous or spectrally bounded.

Keywords

Main Subjects

[1] K. I. Beidar, W. S. Martindale III, A. V. Mikhalev, Rings with generalized identities, Pure and Applied Math., Vol. 196, New York, 1996.
[2] H. E. Bell, W. S. Martindale III, Centeralizing mappings of semiprime rings, Canadian Mathematical Bulletin, 30 (1) (1987), pp. 92-101.
[3] M. Bresar, M. Mathieu, Derivations mapping into the radical III, J. Funct. Anal., 133(1), (1995), pp. 21-29.
[4] C. L. Chung, GPIs having coecients in Utumi quotient rings, proc.Amer.Math.soc., 103 (1988), pp. 723-728.
[5] J. S. Ericson, W. S. Martindale III, J. M. Osborn, Prime nonassociative algebras, pasci c J. math., 60 (1975), pp. 49-63.
[6] B. E. Jacobson, A. M. Sinclair, Continuity of derivations and problem of kaplansky, Amer. J. Math., 90 (1968), pp. 1067-1073.
[7] V. K. Kharchenko, Di erential identity of prime rings, Algebra and Logic., 17 (1978), pp. 155-168.
[8] C. Lanski, An Engel condition with derivation, Proc. Amer. Math. Soc., 118 (1993), pp. 731-734.
[9] T. K. Lee, Semiprime rings with di erential identities, Bull. Inst. Math. Acad. Sinica, 20 (1) (1992), pp. 27-38.
[10] W. S. Martindale III, Prime rings satistying a generalized polynomial identity, J. Algebra., 12 (1969), pp. 576-584.
[11] M. Mathieu, G. J. Murphy, Derivations mapping into the radical, Arch. Math., 57 (5) (1991), pp. 469-474.
[12] M. Mathieu, V. Runde, Derivations mapping into the radical II, Bull. london Math. soc., 24 (5)(1992), pp. 485-487.
[13] E. C. Posner, Derivation in prime rings, Proc. Amer. Math. Soc., 8 (1957), pp. 1093-1100.
[14] K. H. Park, On derivations in non-commutative semiprime rings and Banach algebras, Bull. Korean Math. Soc., 42 (4)(2005), pp. 671-678.
[15] A. M. Sinclair, Continuous derivations on Banach algebras, Proc. Amer. Math. Soc., 20 (1969), pp. 166-170.
[16] I. M. Singer, J. Werner, Derivations on commutative normed algebras, Math. Ann., 129 (1955), pp. 260-264.
[17] M. P. Thomas, The image of a derivation is contained in the radical, math. Ann., 128 (2) (1988), pp. 435-460.