Let [Formula: see text] be an automorphism and[Formula: see text] be a [Formula: see text]-skew [Formula: see text]-derivation of an [Formula: see text]-algebra [Formula: see text]. We prove that if [Formula: see text] is semiprimitive and [Formula: see text] is algebraic, then the subalgebra [Formula: see text] has nilpotent Jacobson radical. Using this result, we obtain similar relations for the Baer prime radical, the Levitzki locally nilpotent radical, and the Köthe nil radical when the field [Formula: see text] is uncountable. Then we apply it to actions of the [Formula: see text]-dimensional Taft Hopf algebra [Formula: see text] and the [Formula: see text]-analogue [Formula: see text] of the enveloping algebra of the Lie algebra [Formula: see text].
The subalgebra A of a Leibniz algebra L is self-idealizing in L, if A = IL (A) . In this paper we study the structure of Leibniz algebras, whose subalgebras are either ideals or self-idealizing. More precisely, we obtain a description of such Leibniz algebras for the cases where the locally nilpotent radical is Abelian non-cyclic, non-Abelian noncyclic, and cyclic of dimension 2.
Abstract
Following J. S. Rose, a subgroup 𝐻 of a group 𝐺 is said to be contranormal in 𝐺 if
G
=
H
G
G=H^{G}
.
It is well known that a finite group is nilpotent if and only if it has no proper contranormal subgroups.
We study nilpotent-by-Černikov groups with no proper contranormal subgroups.
Furthermore, we study the structure of groups with a finite proper contranormal subgroup.
Let [Formula: see text] be a reduced ring containing [Formula: see text] and let [Formula: see text] be commuting locally nilpotent derivations of [Formula: see text]. In this paper, we give an algorithm to decide the local nilpotency of derivations of the form [Formula: see text], where [Formula: see text] are elements in [Formula: see text].
Subalgebra generated by ad-locally nilpotent elements of Borcherds Generalized Kac-Moody Lie algebras
