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.