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On Jordan generalized higher derivations in rings. (English) Zbl 1069.16039
Let $$R$$ be a 2-torsion free ring and $$S\subseteq R$$. A family $$(d_i)_\mathbb{N}$$ of additive maps of $$R$$ is a higher derivation of $$S$$ to $$R$$ if for all $$n\in\mathbb{N}$$ and all $$x,y\in S$$, $$d_n(xy)=\sum_{i+j=n}d_i(x)d_j(y)$$, where $$d_0$$ is the identity map on $$R$$. A second such family $$(f_i)_\mathbb{N}$$ is a generalized higher derivation of $$S$$ to $$R$$ when $$f_n(xy)=\sum_{i+j=n}f_i(x)d_j(y)$$ for all $$n\in\mathbb{N}$$ and all $$x,y\in S$$, and is a Jordan generalized higher derivation (JGHD) of $$S$$ to $$R$$ when one assumes the last equations only for $$y=x$$.
The main result shows that if $$U$$ is a Lie ideal of $$R$$, if $$U$$ is closed under taking squares, and if $$R$$ contains a commutator that is not a right zero divisor, then any JGHD $$(f_i)_\mathbb{N}$$ of $$U$$ to $$R$$ is a generalized higher derivation of $$U$$ to $$R$$.

##### MSC:
 16W25 Derivations, actions of Lie algebras 16W10 Rings with involution; Lie, Jordan and other nonassociative structures 16N60 Prime and semiprime associative rings