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Some results for generalized Lie ideals in prime rings with derivation. II. (English) Zbl 0987.16028

Let \(R\) be a prime ring, \(\text{char }R\neq 2\), \(Z\) the center of \(R\), \(D\) a nonzero derivation of \(R\), and \(\sigma,\tau\colon R\to R\). For \(x,y\in R\) set \([x,y]_{\sigma,\tau}=x\sigma(y)-\tau(y)x\), and for \(S,T\subseteq R\) let \([S,T]_{\sigma,\tau}\) be the additive subgroup of \(R\) generated by \(\{[s,t]_{\sigma,\tau}\mid s\in S\) and \(t\in T\}\). The authors claim the following: if \(a\in R\) so that \([D(R),a]_{\sigma,\tau}=0\) or \(D([R,a]_{\sigma,\tau})=0\) then \(\sigma(a)+\tau(a)\in Z\). In addition if \((x,y)_{\sigma,\tau}=x\sigma(y)+\tau(y)x\) and \(M\) is a nonzero ideal of \(R\), then \(([R,M]_{\sigma,\tau},a)_{\sigma,\tau}= 0\) forces \(a\in Z\). Finally, if \(I=\text{id}_R\) then \((D(R),a)_{I,I}=0\) if and only if \(D((R,a)_{I,I})=0\).

MSC:

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