Multiplicative δ-derivation in alternative algebras

Every multiplicative [Formula: see text]-derivation of an alternative algebra [Formula: see text] is additive if there exists an idempotent [Formula: see text] in [Formula: see text] satisfying the following conditions: (i) [Formula: see text] implies [Formula: see text]; (ii) [Formula: see text] implies [Formula: see text]; (iii) [Formula: see text] implies [Formula: see text] for [Formula: see text]. In particular, every [Formula: see text]-derivation of a prime alternative algebra with a nontrivial idempotent is additive. This generalizes the known result obtained by Rodrigues, Guzzo and Ferreira for [Formula: see text]-derivations. As an application, we apply multiplicative [Formula: see text]-derivation to an alternative complex algebra [Formula: see text] of all [Formula: see text] complex matrices to see how it decomposes into a sum of [Formula: see text]-inner derivation and a [Formula: see text]-derivation on [Formula: see text] given by an additive derivation [Formula: see text] on [Formula: see text].

Footnote to a Formula of Gioia and Subbarao

Recently Gioia and Subbarao [2] studied essentially the following problem: If g(n) is an arithmetic function, and , then what is the behaviour of H(a, n) defined for each fixed integer a ≥ 2 by1By using Vaidyanathaswamy′s formula [e.g., 1], they obtain an explicit formula for H(a, n) in case g(n) is positive and completely multiplicative (Formula 2.2 of [2]). However, Vaidyanathaswamy′s formula is unnecessary to the proof of this result, which indeed follows more simply without its use, by exploiting a simple idea used earlier by Subbarao [3] (referred to also in the course of [2]).