ON (σ, τ )− LEFT JORDAN IDEALS AND GENERALIZED DERIVIATIONS

Abstract

Let R be a prime ring with characteristic not 2 and σ, τ, α, β, λ, μ, γ
automorphisms of R. Let h : R −→ R be a nonzero right (resp. left)-
generalized (α, β)− derivation associated with (α, β)− derivation d ( resp.
d1). Let W, V be nonzero left (σ, τ)−Jordan ideals of R. The main object
in this paper is to study the situations. (1) h(W)=0, (2) [b, W]λ,μ = 0
or [W, b]λ,μ = 0, (3) (b, W)λ,μ = 0 or (W, b)λ,μ = 0, (4) b[W, a]λ,μ = 0 or
[W, a]λ,μb = 0 (5) b(W, a)λ,μ = 0 or (W, a)λ,μb = 0, (6) bW ⊂ Cλ,μ(V ) or
W b ⊂ Cλ,μ(V ). (7) (h(R), b)λ,μa = 0 or a(h(R), b)λ,μ = 0