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

Authors

  • Anwar Faraj Department of Applied Sciences, University of Technology , Baghdad-Iraq
  • Abdulrahman Majeed Department of Mathematics, College of Science, University of Baghdad, ,Baghdad- Iraq https://orcid.org/0000-0001-8534-0749

DOI:

https://doi.org/10.24996/

Keywords:

(θ ,φ ) -derivation, generalized (θ ,φ ) -derivation, , higher derivation, , (U,R) -derivation, Lie ideal, prime ring

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.

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Published

2024-03-04

Issue

Section

Mathematics

How to Cite

ON GENERALIZED(θ ,φ ) -HIGHER DERIVATIONS AND GENERALIZED(U,R) − (θ ,φ ) -HIGHER DERIVATIONS OF PRIME RINGS. (2024). Iraqi Journal of Science, 53(2), 410-418. https://doi.org/10.24996/

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