Binary Lie algebras satisfying the third Engel condition

Let Φ be a unital associative commutative ring with 1/2. The local nilpotency is proved of binary Lie Φ-algebras satisfying the third Engel condition. Moreover, it is proved that this class of algebras does not contain semiprime algebras.     

On Artinian rings satisfying the Engel condition

Let R be an Artinian ring, not necessarily with a unit, and let R ^o be the group of all invertible elements of R with respect to the operation a o b = a + b + ab. We prove that the group R ^o 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 commutative. In particular, R ^o is nilpotent if it is a weakly nilpotent group or an n-Engel group for some positive integer n. We also establish that the ring R is strictly Lie-nilpotent if and only if it is an Engel ring and the quotient ring of the ring R by its Jacobson radical is commutative. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 9, pp. 1264–1270, September, 2006.     

Associative rings satisfying the Engel condition

Let be a commutative ring, and let be an associative -algebra generated by elements . We show that if satisfies the Engel condition of degree , then is upper Lie nilpotent of class bounded by a function that depends only on and . We deduce that the Engel condition in an arbitrary associative ring is inherited by its group of units, and implies a semigroup identity.

     

On locally finite -groups satisfying an Engel condition

For a given positive integer and a given prime number , let be the integer satisfying . We show that every locally finite -group, satisfying the -Engel identity, is (nilpotent of -bounded class)-by-(finite exponent) where the best upper bound for the exponent is either or if is odd. When the best upper bound is or . In the second part of the paper we focus our attention on -Engel groups. With the aid of the results of the first part we show that every -Engel -group is soluble and the derived length is bounded by some constant.

     

Group algebras with units satisfying an Engel identity

We classify group algebras of periodic groups over a field of positive characteristic with units satisfying an Engel identity.     

The nilpotency of Leibniz algebras with Engel condition

It is proved that a Leibniz algebra over a field of zero characteristic with the Engel condition is nilpotent.     

The Gelfand-Kirillov dimension of quadratic algebras satisfying the cyclic condition

We consider algebras over a field presented by generators and subject to square-free relations of the form with every monomial , appearing in one of the relations. It is shown that for 1$"> the Gelfand-Kirillov dimension of such an algebra is at least two if the algebra satisfies the so-called cyclic condition. It is known that this dimension is an integer not exceeding . For , we construct a family of examples of Gelfand-Kirillov dimension two. We prove that an algebra with the cyclic condition with generators has Gelfand-Kirillov dimension if and only if it is of -type, and this occurs if and only if the multiplicative submonoid generated by is cancellative.

     

On the rigid complex associative algebras

The components of the set Alg_n of unitary complex associative laws are known for the dimension n ≥ 5 (see [MZ], [GA]),The problem of finding these components for n ≥ 6 is still open. In this paper we are interested by the determination of the components defined by the rigid algebras, i.e. the algebras whose orbits under the natural GL(n,C)-action are open.     

On pro-unipotent groups satisfying the Golod-Shafarevich condition

We prove that a pro-unipotent group satisfying the Golod-Shafarevich condition contains a free non-abelian pro-unipotent group. Together with the result of A. Magid this implies that such a group is not linear.