Higher-order commutators with power central values on rings and algebras involving generalized derivations

  • 1 Department of Mathematics, King Abdulaziz University, Faculty of Science, Jeddah, Saudi Arabia
  • 2 Department of Mathematics, King Abdulaziz University, Faculty of Science, Jeddah, Saudi Arabia
  • 3 Department of Mathematics, King Abdulaziz University, Faculty of Science & Arts-Rabigh, Jeddah, Saudi Arabia
Shakir Ali
  • Corresponding author
  • Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia, and Department of Mathematics, Faculty of Science, Aligarh Muslim University, Aligarh-202002, India
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, Husain Alhazmi, Abdul Nadim Khan
  • Department of Mathematics, Faculty of Science & Arts-Rabigh, King Abdulaziz University, Jeddah, Saudi Arabia
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and Mohd Arif Raza
  • Department of Mathematics, Faculty of Science & Arts-Rabigh, King Abdulaziz University, Jeddah, Saudi Arabia
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Abstract

Let be a ring with center Z(). In this paper, we study the higher-order commutators with power central values on rings and algebras involving generalized derivations. Motivated by [A. Alahmadi, S. Ali, A. N. Khan and M. Salahuddin Khan, A characterization of generalized derivations on prime rings, Comm. Algebra 44 2016, 8, 3201–3210], we characterize generalized derivations and related maps that satisfy certain differential identities on prime rings. Precisely, we prove that if a prime ring of characteristic different from two admitting generalized derivation 𝔉 such that ([𝔉(sm)sn+sn𝔉(sm),sr]k)lZ() for every s, then either 𝔉(s)=ps for every s or satisfies s4 and 𝔉(s)=sp for every s and p𝔘, the Utumi quotient ring of . As an application, we prove that any spectrally generalized derivation on a semisimple Banach algebra satisfying the above mentioned differential identity must be a left multiplication map.

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