Derivations in semiprime rings and Banach algebras

A note on derivations in semiprime rings

We prove in this note the following result. Let n > 1 be an integer and let R be an n!torsion-free semiprime ring with identity element. Suppose that there exists an additive mapping D : R→ R such that D(xn) =∑nj=1 xn− jD(x)x j−1is fulfilled for all x ∈ R. In this case, D is a derivation. This research is motivated by the work of Bridges and Bergen (1984). Throughout, R will represent an associ...

Generalized Derivations in Prime Rings and Noncommutative Banach Algebras

Let R be a prime ring of characteristic different from 2, C the extended centroid of R, and δ a generalized derivations of R. If [[δ(x), x], δ(x)] = 0 for all x ∈ R then either R is commutative or δ(x) = ax for all x ∈ R and some a ∈ C. We also obtain some related result in case R is a Banach algebra and δ is either continuous or spectrally

On (θ, θ)-Derivations in Semiprime Rings

Lie Ideals and Generalized Derivations in Semiprime Rings

Let R be a 2-torsion free ring and L a Lie ideal of R. An additive mapping F : R ! R is called a generalized derivation on R if there exists a derivation d : R to R such that F(xy) = F(x)y + xd(y) holds for all x y in R. In the present paper we describe the action of generalized derivations satisfying several conditions on Lie ideals of semiprime rings.

Generalized Derivations in Semiprime Gamma Rings

σ-Derivations in Banach algebras