Results 1 to 10 of about 16,516 (281)

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 ...
Евстафьев, Р.Ю.
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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.
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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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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
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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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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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An Engel condition with generalized derivations on multilinear polynomials

Israel Journal of Mathematics, 2007
Let \(R\) be a prime ring with center \(Z(R)\), extended centroid \(C\), nonzero right ideal \(I\), right Utumi quotient ring \(U\), and nonzero generalized derivation \(g\). Set \(f=f(x_1,\dots,x_n)\), a multilinear polynomial over \(C\) in noncommuting indeterminates so that the evaluations in \(R\) satisfy \(f(R^n)\nsubseteq C\). The purpose of this
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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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Annihilators of skew derivations with Engel conditions on prime rings

Czechoslovak Mathematical Journal, 2019
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Pehlivan, Taylan, Albas, Emine
openaire   +3 more sources