Let $\Re$ be a ring with center $Z\left(\Re \right)$. 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 ${\left({[𝔉\left(s^m\right)s^n+s^n𝔉\left(s^m\right),s^r]}_k\right)}^l\in Z\left(\Re \right)$ for every $s\in \Re$, then either $𝔉\left(s\right)=ps$ for every $s\in \Re$ or $\Re$ satisfies $s_4$ and $𝔉\left(s\right)=sp$ for every $s\in \Re$ and $p\in 𝔘$, the Utumi quotient ring of $\Re$. 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.

中文翻译：

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在环和代数上具有幂中心值的高阶换向器，涉及广义导数

让 $\Re$ 与中心同在 $ž（\Re ）$。在本文中，我们研究了在具有广义导数的环和代数上具有幂中心值的高阶换向器。由[A. Alahmadi，S。Ali，A。N. Khan和M. Salahuddin Khan，素环上广义导数的表征，通讯。[Algebra 44 2016，8，3201–3210]，我们刻画了满足导数环上某些微分恒等式的广义导数和相关图。精确地，我们证明如果特征环的质数不同于两个接受广义导数的素数$𝔉$ 这样 ${（{[𝔉（s^{米}）s^{ñ}+s^{ñ}𝔉（s^{米}），s^{\mathrm{[R}}]}_{ķ}）}^{升}\in ž（\Re ）$ 每一个 $s\in \Re$，然后 $𝔉（s）=ps$ 每一个 $s\in \Re$ 要么 $\Re$ 满足 $s_4$ 和 $𝔉（s）=sp$ 每一个 $s\in \Re$ 和 $p\in 𝔘$的Utumi商环 $\Re$。作为一个应用，我们证明在满足上述差分恒等式的半简单Banach代数上的任何频谱广义导数都必须是左乘法映射。