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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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An Engel condition with skew derivations
Monatshefte für Mathematik, 2008The authors extend [\textit{C. Lanski}, Proc. Am. Math. Soc. 118, No. 3, 731-734 (1993; Zbl 0821.16037)] from derivations to skew derivations. Let \(R\) be a prime ring and \(L\) a noncommutative Lie ideal of \(R\). For \(x,y\in R\) set \([x,y]_1=[x,y]=xy-yx\) and when \(n>1\) let \([x,y]_n=[[x,y]_{n-1},y]\).
Chou, Ming-Chu, Liu, Cheng-Kai
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Skew derivations with annihilating Engel conditions
Publicationes Mathematicae Debrecen, 2006Let \(R\) be a noncommutative prime ring. Let \(\sigma\) be an automorphism of \(R\), \(\delta\) be a \(\sigma\)-derivation, and \(a\in R\). The authors prove that if \(a[\delta(x),x]_k=0\) for any \(x\in R\), where \(k\) is a fixed positive integer, then either \(a=0\) or \(\delta=0\), except when \(R=M_2(\text{GF}(2))\).
Chuang, C. L., Chou, M. C., Liu, C. K.
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RINGS SATISFYING GENERALIZED ENGEL CONDITIONS
Journal of Algebra and Its Applications, 2012Let R be an associative ring and let x, y ∈ R. Define the generalized commutators as follows: [x, 0y] = x and [x, ky] = [x, k-1y]y - y[x, k-1y](k = 1, 2, …). In this paper we study some generalized Engel rings, i.e. [Formula: see text]-rings (satisfying [xm(x, y), k(x, y)y] = 0), [Formula: see text]-rings (satisfying [xm(x, y), k(x, y)yn(x, y)] = 0 ...
Ramezan-Nassab, M., Kiani, D.
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