Generalized Derivations on Prime Rings Satisfying Certain Identities
Commun. Korean Math. Soc. Published online December 22, 2020
Radwan Mohammed Al-omary and S Khalid Nauman
Department of mathematics, Ibb university, Ibb, Yemen., Jeddah 21589, Saudi Arabia
Abstract : Let R be a ring with characteristic different from 2. An additive mapping F : R−→Riscalled a generalized derivation on R if there exists a derivation d : R −→ R such that F(xy) = F(x)y+xd(y) holds for all x,y ∈ R. In the present paper, we show that if R is a prime ring satisfying certain identities involving a generalized derivation F associated with a derivation d, then R becomes commutative and in some cases d comes out to be zero (i.e., F becomes a left centralizer). We provide some counter examples to justify that the restrictions imposed in the hypotheses of our theorems are not superfluous.
Keywords : prime rings, derivations and generalized derivations, left centralizer