Lie Triple Derivations on Trivial Extension Algebras

Let U be a unital algebra over a unital commutative ringRandMbe a U-bimodule. A trivial extension algebra U Mis defined as an R-algebra with the usual operations of the R-linear space U ⊕Mand the multiplication defined by (u1,m1)(u2,m2) = (u1u2, u1m2 + m1u2) for all u1, u2 ∈ U, m1,m2 ∈ M. In this article, the structure of Lie triple derivations on U Mare given, and it is shown that under some mild restrictions every Lie triple derivation on U M is of the form δ + τ, where δ : U M → U M is a derivation and τ : U M → Z(U M) is an R-linear mapping.