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Generalized derivations with Engel condition on Lie ideals of prime rings

  • Mohammad Aslam Siddeeque EMAIL logo , Ali Ahmed Abdullah und Nazim Khan
Veröffentlicht/Copyright: 26. Oktober 2022
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Georgian Mathematical Journal
Aus der Zeitschrift Georgian Mathematical Journal Band 30 Heft 1

Abstract

Consider ℜ as a prime ring which is non-commutative in structure with a suitable characteristic. Here, 𝒵 ⁢ ( ℜ ) is the center of ℜ and 𝒬 is the Utumi ring of quotients where 𝒞 is the extended centroid of ℜ . Suppose 𝒫 to be a Lie ideal of ℜ which is non-central. Let 𝒦 be a generalized derivation of ℜ related with derivation μ of ℜ . If 𝒦 satisfies certain typical algebraic identities, then we prove that 𝒦 is either the identity map or the zero map or the scalar map and further information is also drawn on the associated scalar unless ℜ embeds in M 2 ⁢ ( 𝒞 ) , a matrix ring of order 2 × 2 over 𝒞 .

Keywords: Prime ring; generalized derivation; Lie ideal; Utumi ring of quotients; generalized polynomial identities (GPIs); polynomial identities (PIs)
MSC 2010: 16W25; 16N60; 16R50 16R60

References

[1] M. Ashraf, S. A. Pary and M. A. Raza, On generalized derivations in semiprime rings involving anticommutator, Beitr. Algebra Geom. 60 (2019), no. 3, 587–598. 10.1007/s13366-019-00435-0Suche in Google Scholar

[2] K. I. Beidar and M. Brešar, Extended Jacobson density theorem for rings with derivations and automorphisms, Israel J. Math. 122 (2001), 317–346. 10.1007/BF02809906Suche in Google Scholar

[3] K. I. Beidar, W. S. Martindale, III and A. V. Mikhalev, Rings with Generalized Identities, Monogr. Textb. Pure Appl. Math. 196, Marcel Dekker, New York, 1996. Suche in Google Scholar

[4] J. Bergen, I. N. Herstein and J. W. Kerr, Lie ideals and derivations of prime rings, J. Algebra 71 (1981), no. 1, 259–267. 10.1016/0021-8693(81)90120-4Suche in Google Scholar

[5] K. I. Beĭdar, Rings with generalized identities. III (in Russian), Vestnik Moskov. Univ. Ser. I Mat. Mekh. (1978), no. 4, 66–73. Suche in Google Scholar

[6] M.-C. Chou and C.-K. Liu, Annihilators of skew derivations with Engel conditions on Lie ideals, Comm. Algebra 44 (2016), no. 2, 898–911. 10.1080/00927872.2014.990028Suche in Google Scholar

[7] C.-L. Chuang, GPIs having coefficients in Utumi quotient rings, Proc. Amer. Math. Soc. 103 (1988), no. 3, 723–728. 10.1090/S0002-9939-1988-0947646-4Suche in Google Scholar

[8] V. De Filippis and G. Scudo, Strong commutativity and Engel condition preserving maps in prime and semiprime rings, Linear Multilinear Algebra 61 (2013), no. 7, 917–938. 10.1080/03081087.2012.716433Suche in Google Scholar

[9] B. Dhara and V. De Filippis, Engel conditions of generalized derivations on left ideals and Lie ideals in prime rings, Comm. Algebra 48 (2020), no. 1, 154–167. 10.1080/00927872.2019.1635608Suche in Google Scholar

[10] T. S. Erickson, W. S. Martindale, 3rd and J. M. Osborn, Prime nonassociative algebras, Pacific J. Math. 60 (1975), no. 1, 49–63. 10.2140/pjm.1975.60.49Suche in Google Scholar

[11] V. K. Harčenko, Differential identities of prime rings (in Russian), Algebra i Logika 17 (1978), no. 2, 220–238, 242–243. 10.1007/BF01670115Suche in Google Scholar

[12] N. Jacobson, Structure of Rings, Amer. Math. Soc. Colloq. Publ. 37, American Mathematical Society, Providence, 1964. Suche in Google Scholar

[13] C. Lanski, An Engel condition with derivation, Proc. Amer. Math. Soc. 118 (1993), no. 3, 731–734. 10.1090/S0002-9939-1993-1132851-9Suche in Google Scholar

[14] C. Lanski and S. Montgomery, Lie structure of prime rings of characteristic 2, Pacific J. Math. 42 (1972), 117–136. 10.2140/pjm.1972.42.117Suche in Google Scholar

[15] T.-K. Lee, Generalized derivations of left faithful rings, Comm. Algebra 27 (1999), no. 8, 4057–4073. 10.1080/00927879908826682Suche in Google Scholar

[16] C.-K. Liu, Derivations with Engel and annihilator conditions on multilinear polynomials, Comm. Algebra 33 (2005), no. 3, 719–725. 10.1081/AGB-200049880Suche in Google Scholar

[17] W. S. Martindale, III, Prime rings satisfying a generalized polynomial identity, J. Algebra 12 (1969), 576–584. 10.1016/0021-8693(69)90029-5Suche in Google Scholar

[18] E. C. Posner, Prime rings satisfying a polynomial identity, Proc. Amer. Math. Soc. 11 (1960), 180–183. 10.1090/S0002-9939-1960-0111765-5Suche in Google Scholar

[19] R. Prestigiacomo, Generalized derivations preserving Engel condition in prime and semiprime rings, J. Algebra Appl. 20 (2021), no. 3, Paper No. 2150041. 10.1142/S0219498821500419Suche in Google Scholar

[20] T.-L. Wong, Derivations with power-central values on multilinear polynomials, Algebra Colloq. 3 (1996), no. 4, 369–378. 10.1142/S1005386706000344Suche in Google Scholar
Received: 2022-01-04
Revised: 2022-06-08
Accepted: 2022-06-13
Published Online: 2022-10-26
Published in Print: 2023-02-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. On the well-posedness of nonlocal boundary value problems for a class of systems of linear generalized differential equations with singularities
  3. Products of Toeplitz operators with angular symbols
  4. Characterization of Lie-type higher derivations of triangular rings
  5. On classifying map of the integral Krichever–Hoehn formal group law
  6. Demicompactness perturbation in Banach algebras and some stability results
  7. On bicomplex 𝔹ℂ-modules lp 𝕜(𝔹ℂ) and some of their geometric properties
  8. On a class of nonlinear degenerate elliptic equations in weighted Sobolev spaces
  9. Asymptotic behaviors of solutions of a class of time-varying differential equations
  10. Fekete–Szegő problem of strongly α-close-to-convex functions associated with generalized fractional operator
  11. Some properties of Vitali sets
  12. Optimal conditions of solvability of periodic problem for systems of differential equations with argument deviation
  13. Generalized derivations with Engel condition on Lie ideals of prime rings
  14. Attractivity of implicit differential equations with composite fractional derivative
  15. Exponentially fitted difference scheme for singularly perturbed mixed integro-differential equations
https://doi.org/10.1515/gmj-2022-2190
Schlagwörter für diesen Artikel
Prime ring; generalized derivation; Lie ideal; Utumi ring of quotients; generalized polynomial identities (GPIs); polynomial identities (PIs)

Artikel in diesem Heft

  1. Frontmatter
  2. On the well-posedness of nonlocal boundary value problems for a class of systems of linear generalized differential equations with singularities
  3. Products of Toeplitz operators with angular symbols
  4. Characterization of Lie-type higher derivations of triangular rings
  5. On classifying map of the integral Krichever–Hoehn formal group law
  6. Demicompactness perturbation in Banach algebras and some stability results
  7. On bicomplex 𝔹ℂ-modules lp 𝕜(𝔹ℂ) and some of their geometric properties
  8. On a class of nonlinear degenerate elliptic equations in weighted Sobolev spaces
  9. Asymptotic behaviors of solutions of a class of time-varying differential equations
  10. Fekete–Szegő problem of strongly α-close-to-convex functions associated with generalized fractional operator
  11. Some properties of Vitali sets
  12. Optimal conditions of solvability of periodic problem for systems of differential equations with argument deviation
  13. Generalized derivations with Engel condition on Lie ideals of prime rings
  14. Attractivity of implicit differential equations with composite fractional derivative
  15. Exponentially fitted difference scheme for singularly perturbed mixed integro-differential equations
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Heruntergeladen am 7.9.2025 von https://www.degruyterbrill.com/document/doi/10.1515/gmj-2022-2190/pdf?lang=de
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