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Involution Satisfying an Engel Condition
Communications in Algebra, 2016We are given a semiprime ring R with involution *. We show that the following conditions are equivalent. Condition 1: For each element x of R, [x*, x](=x*x − xx*) = 0. Condition 2: There is a fixed natural number N′ such that [x*, xN′] = 0, all elements x of R. Condition 3: There is a fixed natural number N such that for each element x of R, , where dx
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For a set of primes \(\pi\), the concept of Engel conditions for finite groups is extended. For example an element \(g\) is a weakly right \(\pi\)-Engel element if for each \(\pi'\)-element \(x\) in the group \(G\) there is a positive integer \(n\) such that the \((n+1)\)-commutator \([x,g,\dots,g]\) is a \(\pi\)-element.
Fan, Yun, Hai, Jinke
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