The development of the concept of "artinian" for Lie algebras

In paper is considered the development of the concept of "artinian" for Lie algebras. The concept of artinian was introduced for associative rings with the minimality condition. At the same time, it extended to modules and subalgebras. A little later they began to consider Artinian Jordan algebras. For such algebras the role of a one-sided ideal is played by a quadratic ideal or, as N. Djecobson called it, the inner ideal. Artinian for Lie algebras through ideals determined Yu.A. Bakhturin, S.A. Pikhtilkov and V.Рњ. Polyakov. They considered special Artinian Lie algebras. S.A. Pikhtilkov applied Artinian Lie algebras to construct the structural theory of special Lie algebras. Georgia Benkart defined the artinian for Lie algebras through inner ideals. F. Lopez, E. Garcia, G. Lozano explored the concept of the inner ideal applied to artinian with the help of Jordan pairs. The definition of artinian for Lie algebras in this paper is presented in three senses: via subalgebras, ideals, and inner ideals. The relationship established between these definitions is established by the authors earlier. Examples of Artinian Lie algebras are considered. The application of Artinian Lie algebras to the solution of the Mikhalev problem is described: the primary radical of the Artinian Lie algebra is solvable.