Characterizing Jordan n-Derivations of Unital Rings Containing Idempotents

Abstract

Assume that $${\mathcal {R}}$$ is a unital ring containing a nontrivial idempotent. By introducing the concept of Jordan n-derivations (n is any positive integer), it is shown that, under certain conditions, every multiplicative (without any linearity and additivity) Jordan n-derivation $$\delta $$ on $${\mathcal {R}}$$ is additive; and furthermore, $$\delta $$ is a Jordan derivation if the characteristic of $${\mathcal {R}}$$ is not $$n-1$$ and $$\delta $$ is a generalized Jordan derivation if the characteristic of $${\mathcal {R}}$$ is $$n-1$$ $$(n>3)$$ . Based on these results, it turns out that a map on $${\mathcal {R}}$$ is a multiplicative Jordan n-derivation if and only if it is an additive Jordan derivation. As applications, multiplicative Jordan n-derivations on triangular rings, prime rings, matrix algebras, nest algebras and von Neumann algebras are, respectively, completely characterized, which generalize some known related results.