Generalized Derivations on Power Values of Lie Ideals in Prime and Semiprime Rings

Abstract

Let be a 2-torsion free ring and let be a noncentral Lie ideal of , and let : → and : → be two generalized derivations of . We will analyse the structure of in the following cases: (a) is prime and ( ) = ( ) for all ∈ and fixed positive integers ̸ = ; (b) is prime and (( V ) ) = ((V ) ) for all , V ∈ and fixed integers , , , , , ≥ 1; (c) is semiprime and (( V) ) = ((V ) ) for all , V ∈ [ , ] and fixed integer ≥ 1; and (d) is semiprime and (( V) ) = ((V ) ) for all , V ∈ and fixed integer ≥ 1.

Some results on derivations and generalized derivations in rings

“…Generalized derivations have been primarily studied on operator algebras. Therefore any investigation from the algebraic point of view might be interesting (see for example [7]and [8]).…”

A Note on Multiplicative (Generalized) - (α, β) - Reverse Derivations on Left Ideals in Prime Rings

“…Ashraf and Rehman [1] showed that a prime ring R with a nonzero ideal I must be commutative if it admits a derivation d satisfying either of the properties d(xy) + xy ∈ Z(R) or d(xy) − xy ∈ Z(R) for all x, y ∈ R. A number of generalizations of such kind of commutativity ideas found in [4,10]. Recently Rehman et al in [9] introduced a concept of multiplicative (generalized) -(α, β)derivations in rings as follows: A mapping G : R → R (not necessarily additive) is called multiplicative (generalized) -(α, β)derivation if there exists a map (neither necessarily additive or derivation) g : R→ R such that G(xy) = G(x)α(y) + β(x)g(y) for all x, y ∈ R. A mapping G : R → R (not necessarily additive) is called multiplicative (generalized) -(α, β) -reverse derivation if there exists a map (neither necessarily additive or derivation) g : R→ R such that G(xy) = G(y)α(x) + β(y)g(x) for all x, y ∈ R, where α and β are automorphisms on R.…”

Notes on the higher derivations on prime rings

“…Vincenzo De Filippis, Nadeem UR Rehman, and Abu Zaid Ansari [11] proved, let be a 2-torsion free ring and let L be a noncentral Lie ideal of R and let F : R −→ R and G : R −→ R be two generalized derivations of R, they analyze the structure of R in the cases:R is prime and R is semiprime ring after satisfy some conditions. In this notes we gave some results about the higher derivation (HD, for short) and Lie ideal of semiprime ring and prime rings.…”