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\(\sigma\)-Lie ideals with derivations as homomorphisms and anti-homomorphisms. (English) Zbl 1124.16028

Summary: Let \(R\) be a 2-torsion free \(\sigma\)-prime ring, \(U\) a nonzero \(\sigma\)-square closed Lie ideal of \(R\) and \(d\) a derivation of \(R\) which commutes with \(\sigma\). If \(d\) acts as a homomorphism or an anti-homomorphism on \(U\), then either \(d=0\) or \(U\subseteq Z(R)\).

MSC:

16W10 Rings with involution; Lie, Jordan and other nonassociative structures
16N60 Prime and semiprime associative rings
16W25 Derivations, actions of Lie algebras
16U70 Center, normalizer (invariant elements) (associative rings and algebras)
16U80 Generalizations of commutativity (associative rings and algebras)
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