A note on power values of generalized derivation in prime ring and noncommutative Banach algebras

الموضوعات :

Sh. Sahebi ¹ (Department of Mathematics, Islamic Azad University, Central Tehran Branch, PO. Code 14168-94351, Iran)
V. Rahmani ² (Department of Mathematics, Islamic Azad University, Central Tehran Branch, PO. Code 14168-94351, Iran)

تاريخ الإرسال : 07 الأحد , صفر, 1433 تاريخ التأكيد : 07 الأحد , صفر, 1433 تاريخ الإصدار : 08 الخميس , ربيع الثاني, 1433

الکلمات المفتاحية: prime ring, Banach algebras, generalized derivation, Martindale quotient ring,

ملخص المقالة :

Let $R$ be a prime ring with extended centroid $C$, $H$ a generalized derivationof $R$ and $n\geq 1$ a fixed integer. In this paper we study the situations: (1) If $(H(xy))^n =(H(x))^n(H(y))^n$ for all $x,y\in R$; (2) obtain some related result in case $R$ is a noncommutativeBanach algebra and $H$ is continuous or spectrally bounded.

المصادر:

[1] K. I. Beidar, Rings of quotients of semiprime rings, Vestnik Moskovskogo Universiteta. 33(5) (1978), pp. 36-43.

[2] K. I. Beidar, W. S. Martindale III, A. V. Mikhalev, Rings with generalized identities, Pure and Applied Math. Vol. 196, New York, 1996.

[3] H. E. Bell, L. C. Kappe, Rings in which derivations satisfy certain algebraic conditions. Acta Math. Hungar. 53 (3-4) (1989), pp. 339-346.

[4] M. Bresar, A note on derivations, Math. J. Okayama Univ. 32 (1990), pp. 83-88.

[5] L. Carini, A. Giambruno, Lie ideals and nil derivations, Boll. Un. Math. Ital. 6 (1985), pp. 497-503.

[6] C. L. Chuang, GPI's having coecients in Utumi quotient rings, proc. Amer. Math. soc. 103 (1988), pp. 723-728.

[7] B. Felzenszwalb, C. Lanski, On the centralizers of ideals and nil derivations, J. Algebra. 83 (1983), pp. 520-530.

[8] H. Garth Dales, P. Aiena, J. Eschmeier, K. Laursen, G. Willis, Introduction to Banach algebras, operators, and harmonicanalysis, Pablished in the U.S.A by Cambridge University Press, New York, (2003).

[9] A. Giambruno, I. N. Herstein, Derivations with nilpotent values, Rend. Circ. Mat. Palermo. 30(2) (1981), pp. 199-206.

[10] I. N. Herstein, Center like elements in prime rings, J. Alebra. 60 (1979), pp. 567-574.

[11] B. E. Jacobson, A. M. Sinclair, Continuity of derivations and problem of kaplansky, Amer. J. Math. 90 (1968), pp. 1067-1073.

[12] N. Jacobson, Structure of rings, Amer. Math. Soc. Colloq. Pub. 37.Providence, RI: Amer. Math.Soc., (1964).

[13] V. K. Kharchenko, Dierential identity of prime rings, Algebra and Logic. 17 (1978), pp. 155-168.

[14] C. Lanski, Derivation with nilpotent values on Lie ideals, Proc. Amer. Math. Soc. 108 (1990), pp. 31-37.

[15] C. Lanski, An engle condition with derivation, Proc. Amer. Math. Soc. 183(3) (1993), pp. 731-734.

[16] T. K. Lee, Semiprime rings with dierential identities, Bull. Inst. Math. Acad. Sinica. 20(1) (1992), pp. 27-38.

[17] W.S. Martindale III, prime rings satistying a generalized polynomial identity, J.Algebra. 12 (1969), pp. 576-584.

[18] C. M. Ndipingwi, Derivations mapping into the radical , A dissertation submitted to the Faculty of Science University of Johannesburg, (2008).

[19] N. Rehman, On generalized derivations as homomorphisms and anti-homomorphisms, Glas. Mat. III. 39(1) (2004), pp. 27-30.

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

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