Classical radicals and the Martindale centroid of Artin and Noetherian Lie algebras

The work relates to the study of the structure of Lie algebras.
The structure of this work is as follows:
– the first section describes the concepts of classical radicals of Lie algebras, the basic definitions and properties of radicals;
– the second section is devoted to the weakly artinian Lie algebras. The main results are the proof of the local nilpotence property of the prime radical of the weakly artinian Lie algebra and the solution of the A.V. Mikhalev problem;
– the third section deals with the application of the Martindale centroid to the study of the structure of Noetherian special Lie algebras. The main result is a solution to the problem of embedding Any Noetherian semiprime special Lie algebra into algebra of matrices over commutative ring which is the direct sum of fields;
– in the fourth section, the properties of the prime radical of graded $\Omega$-groups are considered.
The research results are reflected in the 10 publications of the author in the period from 2015 to 2018, which were completed during graduate studies at the guidance of doctor of physical-mathematical Sciences, Professor S. A. Pikhtil'kova (02.03.1953–24.12.2015) and candidate of physico-mathematical Sciences, associate Professor O. A. Pikhtilkova.