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Higher-order commutators with power central values on rings and algebras involving generalized derivations
Georgian Mathematical Journal ( IF 0.500 ) Pub Date : 2020-06-11 , DOI: 10.1515/gmj-2020-2064
Shakir Ali; Husain Alhazmi; Abdul Nadim Khan; Mohd Arif Raza

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⁢(sm)⁢sn+sn⁢𝔉⁢(sm),sr]k)l∈Z⁢(ℜ) for every s∈ℜ, then either 𝔉⁢(s)=p⁢s for every s∈ℜ or ℜ satisfies s4 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.
更新日期:2020-07-08

 

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