Results 231 to 240 of about 253,823 (270)

RINGS SATISFYING GENERALIZED ENGEL CONDITIONS

Journal of Algebra and Its Applications, 2012
Let 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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Skew derivations with annihilating Engel conditions

Publicationes Mathematicae Debrecen, 2006
Let \(R\) 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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An Engel condition with skew derivations

Monatshefte für Mathematik, 2008
The 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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Engel Condition and p-nilpotency of Finite Groups

Acta Mathematica Sinica, English Series, 2021
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Wang, Lei, Qiu, Zheng Tian, Qiao, Shou Hong   +2 more
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ON n-ENGEL PAIR SATISFYING CERTAIN CONDITIONS

Journal of Algebra and Its Applications, 2014
Let G be a group and h, g ∈ G. The 2-tuple (h, g) is said to be an n-Engel pair, n ≥ 2, if h = [h,n g], g = [g,n h] and h ≠ 1. In this paper, we prove that if (h, g) is an n-Engel pair, hgh-2gh = ghg and ghg-2hg = hgh, then n = 2k where k = 4 or k ≥ 6. Furthermore, the subgroup generated by {h, g} is determined for k = 4, 6, 7 and 8.
Quek, S. G., Wong, K. B., Wong, P. C.
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Semilocal rings with Engel conditions

Archiv der Mathematik, 2006
The relation between the Engel structure of a semilocal ring and that of its multiplicative group is investigated. Suppose that every local ring whose multiplicative group satisfies an m-Engel condition for some positive integer m is an f (m)-Engel ring for some function f .
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Skew Derivations and Engel Conditions

Communications in Algebra, 2013
It is known that for a nonzero derivation d of a prime ring R, if a nonzero ideal I of R satisfies the Engel-type identity [[…[[d(x k 0 ), x k 1 ], x k 2 ],…], x k n ], then R is commutative. Here we extend this result to a skew derivation of R for a Lie ideal I, which has an immediate corollary that replaces d by an automorphism of R. A related result
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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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