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Hypercentral derivations. (English) Zbl 0805.16035

Define generalized commutators for a ring \(R\) by \([x,y]_ 1 = xy - yx\) and for \(k > 1\) set \([x,y]_ k = [[x,y]_{k - 1},y]\), where \(x,y \in R\). The author proves that if \(R\) has no nonzero nil ideal and has a derivation \(D\) satisfying \([D(x^ n),x^ n]_ k = 0\) for \(k \geq 1\), all \(x\in R\), and \(n = n(x) \geq 1\), then the Utumi quotient ring \(U\) of \(R\) contains a central idempotent \(e\) so that \(D(eU) = 0\) and \((1 - e)U\) is commutative. Furthermore, if \(R\) contains no nonzero nil left ideal, then one can allow \(k = k(x)\). When \(R\) is also a prime ring, then for either assumption about nil ideals, either \(D = 0\) or \(R\) is commutative.

MSC:

16W25 Derivations, actions of Lie algebras
16N60 Prime and semiprime associative rings
16U70 Center, normalizer (invariant elements) (associative rings and algebras)
16U80 Generalizations of commutativity (associative rings and algebras)
16S90 Torsion theories; radicals on module categories (associative algebraic aspects)
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