ON GENERALIZED HOMODERIVATIONS OF PRIME RINGS.

Let A be a ring with its center Z (A). An additive mapping ξ: A → A is called a homoderivation on A if ∀ a, b ∈ A: ξ(ab) = ξ(a)ξ(b) + ξ(a)b + aξ(b). An additive map ψ: A → A is called a generalized homoderivation with associated homoderivation ξ on A if ∀ a, b ∈ A: ψ(ab) = ψ(a)ψ(b) + ψ(a)b + aξ(b). This study examines whether a prime ring A with a generalized homoderivation ψ that fulfils specific algebraic identities is commutative. Precisely, we discuss the following identities: ψ(a)ψ(b) + ab ∈ Z (A), ψ(a)ψ(b) - ab ∈ Z (A), ψ(a)ψ(b) + ab ∈ Z (A), ψ(a)ψ(b) - ab ∈ Z (A), ψ(ab) + ab ∈ Z (A), ψ(ab) - ab ∈ Z (A), ψ(ab) + ba ∈ Z (A), ψ(ab) - ba ∈ Z (A) (∀ a, b ∈ A). Furthermore, examples are given to prove that the restrictions imposed on the hypothesis of the various theorems were not superfluous. [ABSTRACT FROM AUTHOR]