Kaya, Kâzım; Gölbaşi, Öznur; Aydin, Neşet Some results for generalized Lie ideals in prime rings with derivation. II. (English) Zbl 0987.16028 Appl. Math. E-Notes 1, 24-30 (2001). 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 \(D((R,a)_{I,I})=0\). Reviewer: Charles Lanski (Los Angeles) Cited in 4 Documents 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 Keywords:commutativity theorems; prime rings; centers; derivations PDFBibTeX XMLCite \textit{K. Kaya} et al., Appl. Math. E-Notes 1, 24--30 (2001; Zbl 0987.16028) Full Text: EuDML EMIS