On generalized (θ, φ )‎-higher derivations and generalized (U, R)‎−(θ, φ )‎-higher derivations of prime rings

الملخص EN

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 f=(fi)i ϵ N is a generalized (U, R) − (θ, θ) higher derivation of R , then f n (ur)= ∑_(i+J=n)▒〖f i (θ n-i (u))dj (θn-j (r)),〗for all u ∈U,r ∈Rو ى ∈N.