Generalized derivations in prime rings and Banach algebras

Asma Ali; Basudeb Dhara; Shahoor Khan

ArticleOriginal scientific text

Title

Generalized derivations in prime rings and Banach algebras

Authors ¹, ², ¹

Affiliations

Department of Mathematics, Aligarh Muslim University, Aligarh-202002, India
Department of Mathematics, Belda College, Belda, Paschim Medinipur, 721424, India

Abstract

Let R be a prime ring with extended centroid C, F a generalized derivation of R and n ≥ 1, m≥ 1 fixed integers. In this paper we study the situations: 1.

{(F (x \circ y))}^{m} = (x \circ y) ⁿ

for all x,y ∈ I, where I is a nonzero ideal of R; 2. (F(x∘y))ⁿ=(x∘y)ⁿ for all x,y ∈ I, where I is a nonzero right ideal of R. Moreover, we also investigate the situation in semiprime rings and Banach algebras.

Keywords

ENG

prime ring, generalized derivation, extended centroid, Utumi quotient ring

N. Argac and H.G. Inceboz, Derivations of prime and semiprime rings, J. Korean Math. Soc. 46 (2009) 997-1005. doi: 10.4134/JKMS.2009.46.5.997.
M. Ashraf and N. Rehman, On commutativity of rings with derivations, Result. Math. 42 (2002) 3-8. doi: 10.1007/BF03323547.
K.I. Beidar, Rings of quotients of semiprime rings, Vestnik Moskov. Univ. Ser I Math. Meh. (Engl. Transl:. Moscow Univ. Math. Bull.) 33 (1978) 36-42.
M. Brešar and J. Vukman, On left derivations and related mappings, Proc. Amer. Math. Soc. 110 (1990) 7-16. doi: 10.1090/S0002-9939-1990-1028284-3.
M. Brešar, On the distance of the composition of the two derivations to be the generalized derivations, Glasgow Math. J. 33 (1991) 89-93. doi: 10.1017/S0017089500008077.
C.M. Chang, Power central values of derivations on multilinear polynomials, Taiwanese J. Math. 7 (2003) 329-338.
C.L. Chuang, GPIs having coefficients in Utumi quotient rings, Proc. Amer. Math. Soc. 103 (1988) 723-728. doi: 10.1090/S0002-9939-1988-0947646-4.
M.N. Daif and H.E. Bell, Remarks on derivations on semiprime rings, Internat. J. Math. & Math. Sci. 15 (1992) 205-206. doi: 10.1155/S0161171292000255.
V. De Filippis and S. Huang, Generalized derivations on semi prime rings, Bull. Korean Math. Soc. 48 (2011) 1253-1259. doi: 10.4134/BKMS.2011.48.6.1253.
B. Dhara, Remarks on generalized derivations in prime and semiprime rings, Internat. J. Math. & Math. Sc. 2010 (Article ID 646587) 6 pages.
T.S. Erickson, W.S. Martindale III and J.M. Osborn, Prime nonassociative algebras, Pacific J. Math. 60 (1975) 49-63. doi: 10.2140/pjm.1975.60.49.
C. Faith and Y. Utumi, On a new proof of Litoff's theorem, Acta Math. Acad. Sci. Hung. 14 (1963) 369-371. doi: 10.1007/BF01895723.
I.N. Herstein, Topics in ring theory (Univ. of Chicago Press, Chicago, 1969).
S. Huang and B. Davvaz, Generalized derivations of rings and Banach algebras, Comm. Algebra 41 (2013) 1188-1194. doi: 10.1080/00927872.2011.642043.
S. Huang, On generalized derivations of prime and semiprime rings, Taiwanese J. Math. 16 (2012) 771-776.
N. Jacobson, Structure of rings (Amer. Math. Soc. Colloq. Pub. 37. Providence, RI: Amer. Math. Soc., 1964).
B.E. Johnson and A.M. Sinclair, Continuity of derivations and a problem of Kaplansky, Amer. J. Math. 90 (1968) 1067-1073. doi: 10.2307/2373290.
V.K. Kharchenko, Differential identity of prime rings, Algebra and Logic 17 (1978) 155-168. doi: 10.1007/BF01670115.
B. Kim, On the derivations of semiprime rings and noncommutative Banach algebras, Acta Math. Sinica 16 (2000) 21-28. doi: 10.1007/s101149900020.
C. Lanski, An engel condition with derivation, Proc. Amer. Math. Soc. 118 (1993) 731-734. doi: 10.1090/S0002-9939-1993-1132851-9.
T.K. Lee, Semiprime rings with differential identities, Bull. Inst. Math. Acad. Sinica 20 (1992) 27-38.
T.K. Lee, Generalized derivations of left faithful rings, Comm. Algebra 27 (1999) 4057-4073. doi: 10.1080/00927879908826682.
W.S. Martindale III, Prime rings satisfying a generalized polynomial identity, J. Algebra 12 (1969) 576-584. doi: 10.1016/0021-8693(69)90029-5.
M. Mathieu, Properties of the product of two derivations of a C*-algebra, Canad. Math. Bull. 32 (1989) 490-497. doi: 10.4153/CMB-1989-072-4.
M.A. Quadri, M.S. Khan and N. Rehman, Generalized derivations and commutativity of prime rings, Indian J. Pure Appl. Math. 34 (2003) 1393-1396.
A.M. Sinclair, Continuous derivations on Banach algebras, Proc. Amer. Math. Soc. 20 (1969) 166-170. doi: 10.1090/S0002-9939-1969-0233207-X.
I.M. Singer and J. Wermer, Derivations on commutative normed algebras, Math. Ann. 129 (1955) 260-264. doi: 10.1007/BF01362370.
M. Thomas, The image of a derivation is contained in the radical, Ann. Math. 128 (1988) 435-460. doi: 10.2307/1971432.
J. Vukman, On derivations in prime rings and Banach algebras, Proc. Amer. Math. Soc. 116 (1992) 877-884.

Title

Generalized derivations in prime rings and Banach algebras

Affiliations

Abstract

Keywords

Bibliography