Results 211 to 220 of about 27,601 (255)

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
openaire   +2 more sources

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.
openaire   +2 more sources

On the Centralizers of Derivations with Engel Conditions

Communications in Algebra, 2013
Let R be a noncommutative prime ring and d, δ two nonzero derivations of R. If δ([d(x), x] n ) = 0 for all x ∈ R, then char R = 2, d 2 = 0, and δ = αd, where α is in the extended centroid of R. As an application, if char R ≠ 2, then the centralizer of the set {[d(x), x] n  | x ∈ R} in R coincides with the center of R.
Cheng-Kai Liu, Wen-Kwei Shiue
openaire   +1 more source

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
openaire   +1 more source

AN ENGEL CONDITION WITH AUTOMORPHISMS FOR LEFT IDEALS

Journal of Algebra and Its Applications, 2013
Let R be a prime ring and L a nonzero left ideal of R. For x, y ∈ R, we denote [x, y] = xy-yx the commutator of x and y. In this paper, we prove that if R admits a non-identity automorphism σ such that [[…[[σ(xn0), xn1], xn2], …], xnk] = 0 for all x ∈ L, where n0, n1, n2, …, nk are fixed positive integers, then R is commutative.
openaire   +2 more sources

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 ...
openaire   +2 more sources

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
openaire   +2 more sources

Involution Satisfying an Engel Condition

Communications in Algebra, 2016
We 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
openaire   +1 more source

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
openaire   +1 more source

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.
openaire   +1 more source