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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
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