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Generalized Derivations in Rings and Their Applications to Banach Algebra

Author

Listed:
  • Amal S. Alali

    (Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
    These authors contributed equally to this work.)

  • Emine Koç Sögütcü

    (Department of Mathematics, Faculty of Science, Kilis 7 Aralık University, Kilis 79000, Turkey
    These authors also contributed equally to this work.)

  • Zeliha Bedir

    (Department of Mathematics, Faculty of Science, Sivas Cumhuriyet University, Sivas 58140, Turkey
    These authors also contributed equally to this work.)

  • Nadeem ur Rehman

    (Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India
    These authors contributed equally to this work.)

Abstract

Let R be a prime ring, and let F denote a generalized derivation associated with a derivation d of R . Consider I as a nonzero ideal of R , and let m , n , k , l be fixed positive integers. In this study, we explore the behavior of the generalized derivation F within the structures of both prime and semiprime rings that satisfy the functional identity [ F ( η ) , d ( τ ) ] m = η n [ η , τ ] l η k , ∀ , η , τ ∈ I . Furthermore, we extend this investigation to the framework of Banach algebras, analyzing how generalized derivations operate in such algebras. A comparative discussion is also presented to highlight the distinctions and similarities in the behavior of generalized derivations within Banach algebraic settings under the above structural condition.

Suggested Citation

  • Amal S. Alali & Emine Koç Sögütcü & Zeliha Bedir & Nadeem ur Rehman, 2026. "Generalized Derivations in Rings and Their Applications to Banach Algebra," Mathematics, MDPI, vol. 14(2), pages 1-12, January.
    • Handle: RePEc:gam:jmathe:v:14:y:2026:i:2:p:295-:d:1839862
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