Multiplicative δ - Derivations of standard algebras

Every multiplicative δ - derivation of a standard algebra U is additive if there exists an idempotent e′, (e′ ≠ 0, 1) in U satisfying the following conditions: (i) aU = 0 implies a = 0 for e = δ(e); (ii) e′Ua = 0 implies U = 0; (iii) ea(e′′ − 1)U e′(1 − e) implies e′ ae = 0 where e = δ2e′ and e′′ = 2δ(e); (iv) e′a(1 − e′) Ue = 0 where e′=δ(e), e′′ = 2δ(e). In particular, × every δ - derivation of a standard algebra U with a non - trivial idempotent is additive. As an application the concept is applied to multiplicative δ - derivation to a standard complex algebra Mn(C) of all n × n complex matrices to see how it decomposes into a sum of δ - inner derivation and a δ - derivation on Mn(C) given by an additive derivation λ on C.