f-Biderivations and Jordan biderivations of unital algebras with idempotents

Abstract

The notion of [Formula: see text]-derivations was introduced by Beidar and Fong to unify several kinds of linear maps including derivations, Lie derivations and Jordan derivations. In this paper, we introduce the notion of [Formula: see text]-biderivations as a natural “biderivation” counterpart of the notion of “[Formula: see text]-derivations”. We first show, under some conditions, that any [Formula: see text]-biderivation is a Jordan biderivation. Then, we turn to study [Formula: see text]-biderivations of a unital algebra with an idempotent. Our second main result shows, under some conditions, that every Jordan biderivation can be written as a sum of a biderivation, an antibiderivation and an extremal biderivation. As a consequence, we show that every Jordan biderivation on a triangular algebra is a biderivation.