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Let R be a prime ring and L a non-central Lie ideal of R. We aim in this paper to classify generalized derivations F of R satisfying the following algebraic identity: aF(uv)s − γ(uv)tn ∈ Z(R)for all u,v ∈ L, where a ∈ R, γ in C the extended centroid of R and n,s,t are fixed positive integers. Moreover, we extend our analysis to the semiprime ring R and discuss applications such as deriving range inclusion results for continuous generalized derivations on Banach algebras. [ABSTRACT FROM AUTHOR]

Let R be a prime ring and I a nonzero ideal of R. The purpose of this paper is to classify generalized derivations of R satisfying some algebraic identities with power values on I. More precisely, we consider two generalized derivations F and H of R satisfying one of the following identities: (1) aF(x)mH(y)m = xnyn for all x, y ∈ I, (2) (F(x) º H(y))m = (x º y)n for all x, y ∈ I, for two fixed positive integers m ≥ 1, n ≥ 1 and a an element of the extended centroid of R. Finally, as an application, the same identities are studied locally on nonvoid open subsets of a prime Banach algebra. [ABSTRACT FROM AUTHOR]

Let 푅 be a prime ring and 퐿 a non-central Lie ideal of 푅. In this paper, we aim to classify the generalized derivations of 푅 satisfying some algebraic identities with power values on 퐿. Moreover, the same identities are studied locally on a two nonvoid open subsets of a prime Banach algebra. [ABSTRACT FROM AUTHOR]

In this paper, we show that some additive mappings of a Banach algebra are derivations. Moreover, we exhibit some commutativity criteria for a prime Banach algebra admitting a derivation satisfying certain algebraic identities. [ABSTRACT FROM AUTHOR]