ON GENERALIZED(θ ,φ ) -HIGHER DERIVATIONS AND GENERALIZED(U,R) − (θ ,φ ) -HIGHER DERIVATIONS OF PRIME RINGS

Abstract

Let U be a Lie ideal of a 2-torsion free prime ring R and θ ,φ be commuting
endomorphisms of R . In this paper we generalize the main result of M. Ashraf, A.
Khan and C. Heatinger on (θ ,φ ) -higher derivation of prime ring R to generalized
(θ ,φ ) -higher derivation of Lie ideal by introducing the concept of generalized (θ ,φ )
-higher derivation. Under some conditions we prove that a Jordan generalized (θ ,φ ) -
higher derivation of U is either a generalized (θ ,φ ) -higher derivation of U or
U ⊆ Z(R) and every Jordan generalized (θ ,θ ) -higher derivation of R is a
generalized (θ ,θ ) -higher derivation of R . Also, we generalize this result to
generalized (U,R) − (θ ,θ ) -higher derivation by introducing the concepts of
(U,R) − (θ ,φ ) -higher derivation and generalized (U,R) − (θ ,φ ) -higher derivation.
Under some conditions we prove that if i i N F f ∈ = ( ) is a generalized (U,R) − (θ ,θ ) -
higher derivation of R , then
f (ur) f ( (u))d ( n j (r))
j
i j n
n i
n i
−
+ =
= Σ θ − θ , for all u∈U, r ∈R,n∈ N.