On generalized derivations satisfying certain identities

Abstract

Let R be a prime ring with char R ≠ 2 and let d be a generalized derivation on R . We study the generalized derivation d satisfying any of the following identities:

On Multiplicative (Generalized)-Derivations and Central Valued Conditions in Prime Rings

“…for all x, y ∈ I, a non-zero ideal of R. These results have been proved for generalized derivations in [4]. In [1], Albaş proved the following theorem: Let R be a prime ring with char(R) ̸ = 2. If R admits a generalized derivation F with associated derivation…”

Generalized reverse derivations and commutativity of prime rings

“…In [5], the authors extended the above results to generalized derivations. In [2], Albas proved that if R is a prime ring with characteristic different from two and (F, d) is a generalized derivation such that F ([x, y]) ± [F (x), F (y)] = 0 for all x, y ∈ R, then R is commutative or d = 0 or d = −I id , where I id is the identical mapping on R. Hence, it should be interesting to study the commutativity of prime and semiprime rings admitting suitably constrained derivations, generalized derivations and so on.…”