Jordan derivations revisited

Abstract

Let d be a Jordan derivation from a ring $\cal{A}$ into an $\cal{A}$-bimodule $\cal{M}$. Our main result shows that the restriction of d to the ideal of $\cal{A}$ generated by certain higher commutators of $\cal{A}$ is a derivation. This general statement is used for proving that under various additional conditions d must be a derivation on $\cal{A}$. Furthermore, several examples of proper Jordan derivations are given, $C^{\ast}$-algebras admitting proper additive jordan derivations are characterized, and the connections with the related problems on jordan homomorphisms and jordan $\cal{A}$-module homomorphisms are discussed.

Jordan derivations of unital algebras with idempotents

“…Herstein's theorem has been extended to different rings and algebras in various directions (see e.g. [3,4,6,9,16,17,21] and references therein); one might very roughly summarize these results by saying that proper Jordan derivations (i.e. those that are not derivations) from rings and algebras into themselves are rather rare and therefore very special.…”

Additive Jordan derivations of reflexive algebras

“…The converse problem of whether an additive (linear) Jordan derivation is an additive (linear) derivation has received many mathematicians' attention for many years. See [4] and references therein. From the classical result of Bresar [2], we know that each additive Jordan derivation of semiprime algebras is an additive derivation.…”

Functional identities and Jordan σ-derivations

“…is a derivation (see [2,Theorem 4.1] and [5] also). In a recent paper, the author and Lin studied a slight generalized definition concerning (Jordan) derivations.…”