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

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))