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

Shakir Ali; Husain Alhazmi; Abdul Nadim Khan; Mohd Arif Raza

doi:10.1515/gmj-2020-2064

Published by De Gruyter June 11, 2020

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

Shakir Ali , Husain Alhazmi , Abdul Nadim Khan and Mohd Arif Raza

From the journal Georgian Mathematical Journal

https://doi.org/10.1515/gmj-2020-2064

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Abstract

Let ℜ be a ring with center Z ⁢ ( ℜ ) . 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 ⁢ ( s m ) ⁢ s n + s n ⁢ 𝔉 ⁢ ( s m ) , s r ] k ) l ∈ Z ⁢ ( ℜ ) for every s ∈ ℜ , then either 𝔉 ⁢ ( s ) = p ⁢ s for every s ∈ ℜ or ℜ satisfies s 4 and 𝔉 ⁢ ( s ) = s ⁢ p for every s ∈ ℜ and p ∈ 𝔘 , the Utumi quotient ring of ℜ . 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.

Keywords: Prime ring; Utumi quotient ring; Banach algebra; derivation; generalized derivation; spectrally bounded generalized derivation; generalized polynomial identities (GPI); Engel condition

MSC 2010: 16W25; 16N60; 16R50; 46J45

Funding source: King Abdulaziz University

Award Identifier / Grant number: D-050-662-1438

Funding statement: This work was supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant No. D-050-662-1438. The authors, therefore, gratefully acknowledge the DSR technical and financial support.

Acknowledgements

The authors are thankful for the referee’s careful reading of the manuscript.

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Received: 2019-04-25

Accepted: 2019-11-13

Published Online: 2020-06-11

Published in Print: 2021-10-01

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

Abstract

Acknowledgements

References

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