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Remarks on derivations on semiprime rings

Author

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  • Mohamad Nagy Daif
  • Howard E. Bell

Abstract

We prove that a semiprime ring R must be commutative if it admits a derivation d such that (i) x y + d ( x y ) = y x + d ( y x ) for all x , y in R , or (ii) x y − d ( x y ) = y x − d ( y x ) for all x , y in R . In the event that R is prime, (i) or (ii) need only be assumed for all x , y in some nonzero ideal of R .

Suggested Citation

  • Mohamad Nagy Daif & Howard E. Bell, 1992. "Remarks on derivations on semiprime rings," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 15, pages 1-2, January.
    • Handle: RePEc:hin:jijmms:863506
    DOI: 10.1155/S0161171292000255
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    Cited by:

    1. Ali Yahya Hummdi & Öznur Gölbaşı & Emine Koç Sögütcü & Nadeem ur Rehman, 2024. "Some Equations in Rings Involving Semiprime Ideals and Multiplicative Generalized Semiderivations," Mathematics, MDPI, vol. 12(18), pages 1-13, September.
    2. Asma Ali & Inzamam ul Huque, 2020. "Commutativity of a 3-Prime near Ring Satisfying Certain Differential Identities on Jordan Ideals," Mathematics, MDPI, vol. 8(1), pages 1-11, January.
    3. Vincenzo De Filippis & Nadeem UR Rehman & Abu Zaid Ansari, 2014. "Generalized Derivations on Power Values of Lie Ideals in Prime and Semiprime Rings," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2014, pages 1-8, June.
    4. Muhammad Anwar Chaudhry & Öznur Gölbaşi & Emine Koç, 2015. "Some Results on Generalized -Derivations in -Prime Rings," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2015, pages 1-6, April.
    5. Shakir Ali & Turki M. Alsuraiheed & Mohammad Salahuddin Khan & Cihat Abdioglu & Mohammed Ayedh & Naira N. Rafiquee, 2023. "Posner’s Theorem and ∗-Centralizing Derivations on Prime Ideals with Applications," Mathematics, MDPI, vol. 11(14), pages 1-20, July.
    6. Emine Koç Sögütcü & Shuliang Huang, 2023. "Note on Lie ideals with symmetric bi-derivations in semiprime rings," Indian Journal of Pure and Applied Mathematics, Springer, vol. 54(2), pages 608-618, June.
    7. Shakir Ali & Amal S. Alali & Vaishali Varshney, 2024. "Linear Generalized n -Derivations on C ∗ -Algebras," Mathematics, MDPI, vol. 12(10), pages 1-11, May.

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