Lie Triple Derivable Mappings on Rings

Abstract

Let ℛ be a ring containing a nontrivial idempotent. In this article, under a mild condition on ℛ, we prove that if δ is a Lie triple derivable mapping from ℛ into ℛ, then there exists a Z A, B (depending on A and B) in its centre 𝒵(ℛ) such that δ(A + B) = δ(A) + δ(B) + Z A, B . In particular, let ℛ be a prime ring of characteristic not 2 containing a nontrivial idempotent. It is shown that, under some mild conditions on ℛ, if δ is a Lie triple derivable mapping from ℛ into ℛ, then δ = D + τ, where D is an additive derivation from ℛ into its central closure T and τ is a mapping from ℛ into its extended centroid 𝒞 such that τ(A + B) = τ(A) + τ(B) + Z A, B and τ([[A, B], C]) = 0 for all A, B, C ∈ ℛ.