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Remarks on derivations on semiprime rings. (English) Zbl 0746.16029

The main theorem of this short note proves that if \(R\) is a semi-prime ring with a nonzero ideal \(K\) and a derivation \(D\) which satisfies either \(D([x,y])=[x,y]\) for all \(x,y \in K\), or \(D([x,y])=-[x,y]\) for all \(x,y \in K\), then \(K\) is central.

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)
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