Lie triple derivations on primitive rings

In this paper, we show that for each Lie triple derivation L on primitive ring R of characteristic not 2 with nontrivial idempotent, there exists an ordinary derivation D of R into a primitive ring [Formula: see text] containing R and additive mapping λ of R into the center of [Formula: see text] that annihilates commutators such that L(X) = D(X) + λ(X).

Characterization of Multiplicative Lie Triple Derivations on Rings

LetRbe a ring having unit 1. Denote byZRthe center ofR. Assume that the characteristic ofRis not 2 and there is an idempotent elemente∈Rsuch thataRe=0⇒a=0 and aR1-e=0⇒a=0. It is shown that, under some mild conditions, a mapL:R→Ris a multiplicative Lie triple derivation if and only ifLx=δx+hxfor allx∈R, whereδ:R→Ris an additive derivation andh:R→ZRis a map satisfyingha,b,c=0for alla,b,c∈R. As applications, all Lie (triple) derivations on prime rings and von Neumann algebras are characterized, which generalize some known results.

Lie Triple Derivations on Triangular Matrices

Let [Formula: see text] be the algebra of all n × n upper triangular matrices over a commutative unital ring [Formula: see text], and let [Formula: see text] be a 2-torsion free unital [Formula: see text]-bimodule. We show that every Lie triple derivation [Formula: see text] is a sum of a standard Lie derivation and an antiderivation.