Abstract
Let R be a prime ring with Utumi quotient ring U. If R admits a generalized derivation F associated with a derivation d such that \( F([x^my, x]_k)^n-[x^my, x]_k =0\) for all \( x, y\in R \) where \( m\ge 0\) and \( n, k \ge 1\) fixed integers, then R is commutative or \( n=1 \), \(d=0\) and F is an identity map. Moreover, we also examine the case R is a semiprime ring. Finally, we apply the above result to noncommutative Banach algebras.
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Acknowledgments
This paper is supported by Islamic Azad University Central Tehran Branch (IAUCTB). The authors want to thank the authority of IAUCTB for their support to complete this research. Also, the authors would like to thank the referee whose fruitful comments have improved this article.
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Sahebi, S., Rahmani, V. (2016). Generalized Derivations on Rings and Banach Algebras. In: Rizvi, S., Ali, A., Filippis, V. (eds) Algebra and its Applications. Springer Proceedings in Mathematics & Statistics, vol 174. Springer, Singapore. https://doi.org/10.1007/978-981-10-1651-6_6
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DOI: https://doi.org/10.1007/978-981-10-1651-6_6
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