Annihilator of Generalized Derivations with Power Values in Rings and Algebras

Let $\mathcal{F}, \mathcal{G}$ be two generalized derivations of prime ring $\mathcal{R}$ with characteristic different from 2 with associated derivations $d_1$ and $d_2$ respectively. We use the symbols $\mathcal{C}=\mathcal{Z(U)}$ and $\mathcal{U}$ to denote the the extended centroid of $R$ and Utumi ring of quotient of $\mathcal{R}$ respectively. Let $0\neq a \in \mathcal{R}$ and $\mathcal{F}$ and $\mathcal{G}$ satisfy $a\{(\mathcal{F}(xy)+\mathcal{G}(yx))^m-[x,y]^n\}=0$ for all $x, y\in \mathcal{J}$, a nonzero ideal, where $m$ and $n$ are natural numbers. Then either $\mathcal{R}$ is commutative or there exists $c$, $b\in \mathcal{U}$ such that $\mathcal{F}$(x) = cx and $\mathcal{G}$(x) = bx for all x ∈ R.

Keywords:

Semiprime rings, Generalized derivations extended centroid,

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[1] Beidar, K. I.: Rings of quotients of semiprime rings. Vestnik Moskov. Univ. Ser I Math. Meh. (Engl. Transl:. Moscow Univ. Math. Bull.). 33,36-42 (1978).
[2] Brešar, M.: On the distance of the composition of the two derivations to be the generalized derivations. Glasgow Math. J. 33 (1), 89-93 (1991).
[3] Chuang, C. L.: GPIs having coefficients in Utumi quotient rings. Proc. Amer. Math. Soc. 103, 723-728 (1988).
[4] Daif, M. N. and Bell, H. E: Remarks on derivations on semiprime rings. Int. J. Math. & Math. Sci. 15 (1) , 205-206 (1992).
[5] Filippis, V. De: Generalized derivations in prime rings and noncommutative Banach algebras. Bull. Korean Math. Soc. 45, 621-629 (2008).
[6] Filippis, V. De and Huang, S.: Generalized derivations on semi prime rings. Bull. Korean Math. Soc. 48 (6), 1253-1259 (2011).
[7] Dhara, B.: Remarks on generalized derivations in prime and semiprime rings. Int. J. Math. & Math. Sc. 2010, Article ID 646587, 6 pages.
[8] Kharchenko, V. K.: Differential identity of prime rings. Algebra and Logic. 17, 155-168 (1978).
[9] Lee, T. K.: Semiprime rings with differential identities. Bull. Inst. Math. Acad. Sinica, 20 (1) , 27-38 (1992).
[10] Lee, T. K.: Generalized derivations of left faithful rings. Comm. Algebra, 27 (8), 4057-4073 (1999).
[11] Park, K. H.: On derivations in noncommutative semiprime rings and Banach algebras. Bull. Korean Math. Soc. 42, 671-678 (2005).
[12] Quadri, M. A., Khan, M. S. and Rehman, N.: Generalized derivations and commutativity of prime rings. Indian J. Pure Appl. Math. 34 (9),1393-1396 (2003).
[13] Sinclair, A. M.: Continuous derivations on Banach algebras. Proc. Amer. Math. Soc. 20, 166-170 (1969).
[14] Singer, I. M. and Wermer, J.: Derivations on commutative normed algebras. Math. Ann. 129, 260–264 (1955).
[15] Johnson, B. E. and Sinclair, A. M.: Continuity of derivations and a problem of Kaplansky. Amer. J. Math. 90, 1067-1073 (1968).
[16] Chuang, C. L. and Lee, T. K.: Rings with annihilator conditions on multilinear polynomials. Chinese J. Math. 177-185 (1996).
[17] Erickson, T. S. and Martindale III, W. S. and Osborn, J. M.: Prime nonassociative algebras. Pacific J. Math. 60, 49-63 (1975).
[18] Martindale III, W. S.: Prime rings satisfying a generalized polynomial identity. J. Algebra. 12, 576-584 (1969).
[19] Jacobson, N.: Structure of rings. Amer. Math. Soc. Colloq. Pub., 37, Amer. Math. Soc., Providence, RI, 1964.
[20] Bell, H. E.and Martindale III, W. S.: Centralizing mappings of semiprime rings. Canad. Math. Bull. 30, 92-101 (1987).

Mathematical Sciences and Applications E-Notes-Cover

ISSN: 2147-6268
Yayın Aralığı: Yılda 4 Sayı
Başlangıç: 2013
Yayıncı: -

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