Results 161 to 170 of about 7,751,448 (200)

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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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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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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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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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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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
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
Pehlivan, Taylan, Albas, Emine
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Annihilators of derivations with Engel conditions on lie ideals

Rendiconti del Circolo Matematico di Palermo, 2003
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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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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