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Commutators with power central values on a Lie ideal. (English) Zbl 1009.16034
Summary: Let \(R\) be a prime ring of characteristic \(\neq 2\) with a derivation \(d\neq 0\), \(L\) a noncentral Lie ideal of \(R\) such that \([d(u),u]^n\) is central, for all \(u\in L\). We prove that \(R\) must satisfy \(s_4\), the standard identity in \(4\) variables. We also examine the case \(R\) is a 2-torsion free semiprime ring and \([d([x,y]),[x,y]]^n\) is central, for all \(x,y\in R\).

MSC:
16W10 Rings with involution; Lie, Jordan and other nonassociative structures
16N60 Prime and semiprime associative rings
16R50 Other kinds of identities (generalized polynomial, rational, involution)
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