ON GENERALIZED(θ ,φ ) -HIGHER DERIVATIONS AND GENERALIZED(U,R) − (θ ,φ ) -HIGHER DERIVATIONS OF PRIME RINGS - Faraj, Anwar; Majeed, Abdulrahman (Iraqi Journal of Science, 2024)
Let U be a Lie ideal of a 2-torsion free prime ring R and θ ,φ be commutingendomorphisms 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 orU ⊆ Z(R) and every Jordan generalized (θ ,θ ) -higher derivation of R is ageneralized (θ ,θ ) -higher derivation of R . Also, we generalize this result togeneralized (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 , thenf (ur) f ( (u))d ( n j (r))ji j nn in i−+ == Σ θ − θ , for all u∈U, r ∈R,n∈ N.