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An Engel condition with derivation. (English) Zbl 0821.16037
Let \(R\) be an associative prime ring and \(d\) a nonzero derivation of \(R\). The author proves the following two generalizations of Posner’s theorem: i) If \(I\) is a nonzero ideal of \(R\) and there exists an integer \(k>0\) such that \([d(r),\underbrace{r,r,\dots,r}_k]=0\) for all \(r\in I\), then \(R\) is commutative; ii) If \(I\) is a noncommutative Lie ideal of \(R\) and there exists an integer \(k>0\) such that \([d(r),\underbrace{r,\dots,r}_k]=0\) for all \(r\in I\), then \(R\subseteq M_2(F)\) where \(F\) is a field of characteristic two.

MSC:
16W25 Derivations, actions of Lie algebras
16W10 Rings with involution; Lie, Jordan and other nonassociative structures
16U80 Generalizations of commutativity (associative rings and algebras)
16N60 Prime and semiprime associative rings
16U70 Center, normalizer (invariant elements) (associative rings and algebras)
16R50 Other kinds of identities (generalized polynomial, rational, involution)
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