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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 + sn𝔉(sm), sr]k)l ∈ Z(ℜ) or every s ∈ ℜ, then either 𝔉(s) = ps for every s ∈ ℜ or ℜ satisfies s4 and 𝔉(s) = sp 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. [ABSTRACT FROM AUTHOR]

Let 1 < k and m, k 2 Z+. In this manuscript, we analyse the action of (semi)-prime rings satisfying certain differential identities on some suitable subset of rings. To be more specific, we discuss the behaviour of the semiprime ring R satisfying the differential identities ([d([s, t]m), [s, t]m])k = [d([s, t]m), [s, t]m] for every s, t 2 R. [ABSTRACT FROM AUTHOR]

Let A be unital prime Banach algebra over ℝ or ℂ with centre and G1, G2 be open subsets of be a continuous linear generalized skew derivation, and be a continuous linear map. We prove that must be commutative if one of the following conditions holds: For each a ∈ G1, b ∈ G2, there exists an integer m ∈ Z>1 depending on a and b such that either. For each a ∈ G1, b ∈ G2, there exists an integer m ∈ Z>1 depending on a and b such that either. These results generalize a number of theorems of this type. In particular, as an application, we give an affirmative answer to some questions posed in [21]. [ABSTRACT FROM AUTHOR]