Descending chain conditions for Lie algebras of prime characteristic

Abstract

The theme of this paper is the study of descending chain conditions for subideals of infinite-dimensional Lie algebras of prime characteristic, complementing the results of the second author [8] on Lie algebras of characteristic zero. Robinson [5] proved that every group satisfying the minimal condition for subnormal subgroups is a finite extension of a Z-group, where a Z-group is a group in which normality is transitive. As a corollary to the proof it follows that every group satisfying the minimal condition for 2-step subnormal subgroups must in addition satisfy the minimal condition for all subnormal subgroups. Lie algebra analogues of these and related results are proved in [8] for fields of characteristic zero. The proofs use properties of the Baer radical of a Lie algebra (Hartley [3]) which are not valid in prime characteristic. For about five years in question has remained open, whether the analogy extends to Lie algebras of prime characteristic: the best that was known until recently was that the minimal condition for 3-step subideals implies that for all subideals. We answer the question here, as follows. Over fields of prime characteristic it remains true that every Lie algebra satisfying the minimal condition for subideals is a finite-dimensional extension of a IL-algebra. However, it is not true that the minimal condition for 2-step subideals implies that for all subideals. In fact a structural analysis of Lie algebras with the minimal