integration as a generalization of the integral operator

Let a be an algebra. a derivation on a is a linear mapping δ : a → a such that δ(ab) = δ(a)b+aδ(b) for every a, b ∈ a. as a dual to this notion, we consider a linear mapping δ : a → a with the property δ(a)δ(b) = δ(δ(a)b + aδ(b)) for every a, b ∈ a and we call it an integration. in this paper, we give some examples, counterexamples and facts concerning integrations on algebras. furthermore, we state and prove a characterization for integrations on finite dimensional matrix algebras.