Characterizing Jordan \(n\)-derivations of unital rings containing idempotents (Q826572)

Let \(R\) be a ring. For \(x,y\in R\) write \(\{x,y\}=xy+yx\). Further, define \(p_1(x)=x\) and \(p_n(x_1,\dots,x_n)= \{p_{n-1}(x_1,\dots,x_{n-1}),x_n\}\) for \(n\ge 2\). The authors call a not necessarily additive map \(\delta:R\to R\) a Jordan \(n\)-derivation if it satisfies \[\delta(p_n(x_1,\dots,x_n)) = \sum_{i=1}^n p_n(x_1,\dots,x_{i-1}, \delta(x_i),x_{i+1},\dots,x_n)\] for all \(x_i\in R\). It is shown that \(\delta\) is necessarily additive if \(R\) contains an idempotent that satisfies certain technical conditions. Moreover, under slightly stronger assumptions, \(\delta\) is a Jordan derivation.