Engel Conditions Research Articles

Engel conditions and symmetric tensors

In a recent study of Engel Lie rings, Serena Cicalò and Willem de Graaf have given a practical set of conditions for an additively finitely generated Lie ring L to satisfy an Engel condition. We present a simpler and more direct proof of this fact. Our main result generalizes this in the language of tensor algebra, and describes a relatively small generating set for the module generated by all n-th tensor powers of elements of a finitely generated ℤ-module M, in terms of a generating set for M.

Identities with Engel conditions on derivations

Let R be a semiprime ring with a derivation D. The focus is on the two identities with Engel condition on $${D: [x^m, D(x^{n_1}),\ldots,D(x^{n_s})]_s=0}$$ for all $${x\in R}$$ and $${[x^m, D(x)^{n_1},\ldots,D(x)^{n_s}]_s=0}$$ for all $${x\in R}$$ , where s, m, n 1, . . . , n s are fixed positive integers. Our results are natural generalizations of Posner’s theorem on centralizing derivations, Herstein’s theorem on derivations with power-central values and a recent result by A. Fošner, M. Fošner and Vukman.

Derivations with Engel conditions on multilinear polynomials in prime rings

AbstractLet

Generalized Derivations with Engel Conditions on One-Sided Ideals

Let R be a noncommutative prime ring and I a nonzero left ideal of R. Let g be a generalized derivation of R such that [g(r k ), r k ] n = 0 for all r ∈ I, where k, n are fixed positive integers. Then there exists c ∈ U, the left Utumi quotient ring of R, such that g(x) = xc and I(c − α) = 0 for a suitable α ∈ C. In particular we have that g(x) = α x, for all x ∈ I.

Semilocal rings with Engel conditions

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 . It is proved that under this condition a corresponding statement holds for every semilocal ring which is generated by its multiplicative group.

Skew derivations with annihilating Engel conditions

Skew derivations with annihilating Engel conditions

4-ENGEL GROUPS ARE LOCALLY NILPOTENT

Questions about nilpotency of groups satisfying Engel conditions have been considered since 1936, when Zorn proved that finite Engel groups are nilpotent. We prove that 4-Engel groups are locally nilpotent. Our proof makes substantial use of both hand and machine calculations.

Annihilators of derivations with Engel conditions on lie ideals

LetR a prime ring,L a noncentral Lie ideal ofR anda ∈R. Suppose thatd is a nonzero derivation ofR such thata[d(u),u] k =0 for allu ∈L, wherek is a fixed positive integer. Thena=0 except when charR=2 and dim C RC=4.

Annihilators of derivations with Engel conditions on one-sided ideals

Annihilators of derivations with Engel conditions on one-sided ideals

A Note on ?-Engel Conditions

In this paper we show that the weakly π’-Engel conditions are closely related to the existance of normal π-complements; while the π’-Engel conditions are closely related to the π-nilpotent groups.

Radical Rings with Engel Conditions

An associative ring R without unity is called radical if it coincides with its Jacobson radical, which means that the set of all elements of R forms a group denoted by R∘ under the circle operation r∘s=r+s+rs on R. It is proved that, for a radical ring R, the group R∘ satisfies an n-Engel condition for some positive integer n if and only if R is m-Engel as a Lie ring for some positive integer m depending only on n.

Some Engel conditions on infinite subsets of certain groups

Let k be a positive integer. We denote by ɛk(∞) the class of all groups in which every infinite subset contains two distinct elements x, y such that [x,k y] = 1. We say that a group G is an -group provided that whenever X, Y are infinite subsets of G, there exists x ∈ X, y ∈ Y such that [x,k y] = 1. Here we prove that:(1) If G is a finitely generated soluble group, then G ∈ ɛ3(∞) if and only if G is finite by a nilpotent group in which every two generator subgroup is nilpotent of class at most 3.(2) If G is a finitely generated metabelian group, then G ∈ ɛk(∞) if and only if G/Zk (G) is finite, where Zk (G) is the (k + 1)-th term of the upper central series of G.(3) If G is a finitely generated soluble ɛk(∞)-group, then there exists a positive integer t depending only on k such that G/Zt (G) is finite.(4) If G is an infinite -group in which every non-trivial finitely generated subgroup has a non-trivial finite quotient, then G is k-Engel. In particular, G is locally nilpotent.

Derivations with Engel conditions on multilinear polynomials

Let R R be a prime algebra over a commutative ring K K with unity and let f ( X 1 , … , X n ) f(X_{1}, \ldots , X_{n}) be a multilinear polynomial over K K . Suppose that d d is a nonzero derivation on R R such that for all r 1 , … , r n r_{1}, \ldots , r_{n} in some nonzero ideal I I of R R , [ d ( f ( r 1 , … , r n ) ) , f ( r 1 , … , r n ) ] k = 0 \Big [ d\big ( f(r_{1}, \ldots , r_{n})\big ), f(r_{1}, \ldots , r_{n}) \Big ]_{k} = 0 with k k fixed. Then f ( X 1 , … , X n ) f(X_{1}, \ldots , X_{n}) is central–valued on R R except when char R = 2 R=2 and R R satisfies the standard identity s 4 s_{4} in 4 variables.

Formations of metanilpotent groups and generalized Engel conditions

Formations of metanilpotent groups and generalized Engel conditions

A characterization of finite p-soluble groups of p-length one by commutator identities

AbstractA classical result of M. Zorn states that a finite group is nilpotent if and only if it satisfies an Engel condition. If this is the case, it satisfies almost all Engel conditions. We shall give a similar description of the class of p-soluble groups of p-length one by a sequence of commutator identities.

Engel conditions and direct decompositions into groups of coprime order

Engel conditions and direct decompositions into groups of coprime order

Closure properties and partial Engel conditions in groups

Closure properties and partial Engel conditions in groups