Generalized derivations in prime rings and Banach algebras

Asma Ali; Basudeb Dhara; Shahoor Khan

Discussiones Mathematicae - General Algebra and Applications (2014)

  • Volume: 34, Issue: 1, page 125-138
  • ISSN: 1509-9415

Abstract

top
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 ∘ y ) ) m = ( x ∘ 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.

How to cite

top

Asma Ali, Basudeb Dhara, and Shahoor Khan. "Generalized derivations in prime rings and Banach algebras." Discussiones Mathematicae - General Algebra and Applications 34.1 (2014): 125-138. <http://eudml.org/doc/270601>.

@article{AsmaAli2014,
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. $(F(x∘y))^m = (x∘y)ⁿ$ 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.},
author = {Asma Ali, Basudeb Dhara, Shahoor Khan},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {prime ring; generalized derivation; extended centroid; Utumi quotient ring},
language = {eng},
number = {1},
pages = {125-138},
title = {Generalized derivations in prime rings and Banach algebras},
url = {http://eudml.org/doc/270601},
volume = {34},
year = {2014},
}

TY - JOUR
AU - Asma Ali
AU - Basudeb Dhara
AU - Shahoor Khan
TI - Generalized derivations in prime rings and Banach algebras
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2014
VL - 34
IS - 1
SP - 125
EP - 138
AB - 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. $(F(x∘y))^m = (x∘y)ⁿ$ 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.
LA - eng
KW - prime ring; generalized derivation; extended centroid; Utumi quotient ring
UR - http://eudml.org/doc/270601
ER -

References

top
  1. [1] 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. Zbl1185.16044
  2. [2] M. Ashraf and N. Rehman, On commutativity of rings with derivations, Result. Math. 42 (2002) 3-8. doi: 10.1007/BF03323547. Zbl1038.16021
  3. [3] 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. Zbl0403.16003
  4. [4] 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. Zbl0703.16020
  5. [5] 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. Zbl0731.47037
  6. [6] C.M. Chang, Power central values of derivations on multilinear polynomials, Taiwanese J. Math. 7 (2003) 329-338. Zbl1058.16032
  7. [7] 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. Zbl0656.16006
  8. [8] 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. Zbl0746.16029
  9. [9] 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. Zbl1232.16030
  10. [10] B. Dhara, Remarks on generalized derivations in prime and semiprime rings, Internat. J. Math. & Math. Sc. 2010 (Article ID 646587) 6 pages. Zbl1220.16030
  11. [11] 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. Zbl0355.17005
  12. [12] 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. Zbl0147.28602
  13. [13] I.N. Herstein, Topics in ring theory (Univ. of Chicago Press, Chicago, 1969). Zbl0232.16001
  14. [14] S. Huang and B. Davvaz, Generalized derivations of rings and Banach algebras, Comm. Algebra 41 (2013) 1188-1194. doi: 10.1080/00927872.2011.642043. 
  15. [15] S. Huang, On generalized derivations of prime and semiprime rings, Taiwanese J. Math. 16 (2012) 771-776. Zbl1252.16038
  16. [16] N. Jacobson, Structure of rings (Amer. Math. Soc. Colloq. Pub. 37. Providence, RI: Amer. Math. Soc., 1964). 
  17. [17] 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. Zbl0179.18103
  18. [18] V.K. Kharchenko, Differential identity of prime rings, Algebra and Logic 17 (1978) 155-168. doi: 10.1007/BF01670115. Zbl0423.16011
  19. [19] B. Kim, On the derivations of semiprime rings and noncommutative Banach algebras, Acta Math. Sinica 16 (2000) 21-28. doi: 10.1007/s101149900020. Zbl0973.16020
  20. [20] C. Lanski, An engel condition with derivation, Proc. Amer. Math. Soc. 118 (1993) 731-734. doi: 10.1090/S0002-9939-1993-1132851-9. Zbl0821.16037
  21. [21] T.K. Lee, Semiprime rings with differential identities, Bull. Inst. Math. Acad. Sinica 20 (1992) 27-38. Zbl0769.16017
  22. [22] T.K. Lee, Generalized derivations of left faithful rings, Comm. Algebra 27 (1999) 4057-4073. doi: 10.1080/00927879908826682. Zbl0946.16026
  23. [23] 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. Zbl0175.03102
  24. [24] 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. Zbl0641.46037
  25. [25] 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. Zbl1047.16020
  26. [26] A.M. Sinclair, Continuous derivations on Banach algebras, Proc. Amer. Math. Soc. 20 (1969) 166-170. doi: 10.1090/S0002-9939-1969-0233207-X. 
  27. [27] I.M. Singer and J. Wermer, Derivations on commutative normed algebras, Math. Ann. 129 (1955) 260-264. doi: 10.1007/BF01362370. Zbl0067.35101
  28. [28] M. Thomas, The image of a derivation is contained in the radical, Ann. Math. 128 (1988) 435-460. doi: 10.2307/1971432. Zbl0681.47016
  29. [29] J. Vukman, On derivations in prime rings and Banach algebras, Proc. Amer. Math. Soc. 116 (1992) 877-884. Zbl0792.16034

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.