Multiplicative Delta-Derivation In Alternative Algebras

Every multiplicative delta-derivation of an alternative algebra A is additive if there exists an idempotent e' (e' not equal 0, 1) in A satisfying the following conditions: (i) e' Au = 0 implies A = 0; (ii) e'ueA(1 - e) = 0 implies e'ue = 0; (iii) uA = 0 implies u = 0 for e' = delta(e). In particular, every delta-derivation of a prime alternative algebra with a nontrivial idempotent is additive. This generalizes the known result obtained by Rodrigues, Guzzo and Ferreira for delta-derivations. As an application, we apply multiplicative delta-derivation to an alternative complex algebra M-n(C) of all n x n complex matrices to see how it decomposes into a sum of delta-inner derivation and a delta-derivation on M-n(C) given by an additive derivation gamma on C.