Results 121 to 130 of about 174,282 (159)

Power closure and the Engel condition

Israel Journal of Mathematics, 1997
A Lie \(p\)-algebra \(L\) is called \(n\)-power closed if in every section of \(L\), any sum of two \(p^{i+n}\)-th powers is a \(p^i\)-th power \((i>0)\). The authors prove that if \(L\) is residually nilpotent and \(n\)-power closed for some \(n\geq 0\) then \(L\) is \((3p^{n+2}+1)\)-Engel if \(p\geq 2\) and \((3\cdot 2^{n+3}+1)\)-Engel if \(p=2 ...
Riley, David M., Semple, James F.
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On Artinian rings satisfying the Engel condition

Ukrainian Mathematical Journal, 2006
Summary: Let \(R\) be an Artinian ring, not necessarily with a unit, and let \(R^\circ\) be the group of all invertible elements of \(R\) with respect to the operation \(a\circ b=a+b+ab\). We prove that the group \(R^\circ\) is a nilpotent group if and only if it is an Engel group and the quotient ring of the ring \(R\) by its Jacobson radical is ...
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Generalized derivations with Engel condition on multilinear polynomials

Israel Journal of Mathematics, 2009
Let \(R\) be a prime ring with right Utumi quotient ring \(U\), extended centroid \(C\), nonzero right ideal \(I\), and nonzero generalized derivation \(D\). For \(x,y\in R\) let \(xy-yx=[x,y]=[x,y]_1\) and for \(k>1\) set \([x,y]_k=[[x,y]_{k-1},y]\). The main result in the paper assumes that \([D(f(a_1,\dots,a_n)),f(a_1,\dots,a_n)]_k=0\) for a nonzero
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Orderable Groups Satisfying an Engel Condition

1993
The purpose of this note is to point out that lattice-ordered groups satisfying bounded Engel condition are nilpotent. Several similar results are also obtained for orderable groups. The techniques used come from recent studies in residually finite p-groups and from results of Zel’manov for Engel groups.
Y. K. Kim, A. H. Rhemtulla
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On Certain Weak Engel-Type Conditions in Groups

Communications in Algebra, 2014
Let w(x, y) be a word in two variables and 𝔚 the variety determined by w. In this paper we raise the following question: if for every pair of elements a, b in a group G there exists g ∈ G such that w(a g , b) = 1, under what conditions does the group G belong to 𝔚? In particular, we consider the n-Engel word w(x, y) = [x, n y].
MERIANO, MAURIZIO, NICOTERA, Chiara
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Anti-automorphisms satisfying an Engel condition

Communications in Algebra, 2016
ABSTRACTLet R be a semiprime ring with an anti-automorphism τ, which is of finite order. It is proved that if [[…[τ(x),xn1],…],xnk]=0 for all x∈R, where n1,n2,…,nk are k fixed positive integers, then τ is a commuting map. Moreover, commuting anti-automorphisms of semiprime rings are also characterized.
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Lie rings satisfying the Engel condition

Mathematical Proceedings of the Cambridge Philosophical Society, 1954
1. Let be a Lie ring in which the product of elements x and y is denoted by xy. The inner derivations of , i.e. the mappings X:a→ax for fixed elements x of , form a Lie ring under the product [X, Y] = XY – YX, and the mapping x→ X is a homo-morphism of onto . We shall say that satisfies the nth Engel condition if Xn = 0 for all X in , i.e. iffor all
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Some skew linear groups with Engel's condition

Journal of Group Theory, 2012
Abstract ...
Mojtaba Ramezan-Nassab, Dariush Kiani
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Antiautomorphisms with quasi-generalized Engel condition

Journal of Algebra and Its Applications, 2018
Let [Formula: see text] be a ring with 1. Given elements [Formula: see text], [Formula: see text] of [Formula: see text] and the integer [Formula: see text] define [Formula: see text] and [Formula: see text]. We say that a given antiautomorphism [Formula: see text] of [Formula: see text] is commuting if [Formula: see text], all [Formula: see text ...
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On the additive maps satisfying skew-Engel conditions

2017
Summary: Let \(R\) be a prime ring, \(I\) be any nonzero ideal of \(R\) and \(f:I\to R\) be an additive map. Then skew-Engel condition \(\langle\ldots\langle\langle f(x),x^{n_1}\rangle,x^{n_2}\rangle,\ldots, x^{n_k}\rangle=0\) implies that \(f(x)=0\;\forall x\in I\) provided \(2\neq \operatorname{char}(R) > n_1 + n_2 + \ldots + n_k\), where \(n_1,n_2 ...
Nadeem, M., Aslam, M., Ahmed, Y.
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