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On derivations in \(\sigma\)-prime rings. (English) Zbl 1124.16025

Summary: Let \((R,\sigma)\) be a 2-torsion free \(\sigma\)-prime ring with involution \(\sigma\) and \(d\) be a nonzero derivation of \(R\). In the present paper it is shown that if \(d\) is centralizing, then \(R\) is commutative. Furthermore, if \(d\) commutes with \(\sigma\) and \(0\neq I\) is a \(\sigma\)-ideal of \(R\) such that either \([d(x),d(y)]=0\) or \(d(xy)=d(yx)\) for all \(x,y\in I\), then \(R\) is a commutative ring.

MSC:

16W10 Rings with involution; Lie, Jordan and other nonassociative structures
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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