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An identity on partial generalized automorphisms of prime rings. (English) Zbl 1282.16042
Let \(R\) be a prime ring, let \(L\) be a Lie ideal of \(R\), and let \(T\) be an automorphism of \(R\). Suppose there exist integers \(s\geq 0\), \(t\geq 0\), \(n\geq 1\) such that \((u^s(T(u)u+uT(u))u^t)^n=0\) for all \(u\in L\). It is shown that if \(\text{char}(R)=0\) or \(\text{char}(R)>n+1\), then \(L\) is central. (The result is actually stated for the so-called partial generalized automorphisms, but in the context of this problem these are just automorphisms multiplied by an element from the extended centroid.)
MSC:
16W20 Automorphisms and endomorphisms
16N60 Prime and semiprime associative rings
16W10 Rings with involution; Lie, Jordan and other nonassociative structures
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