Generalised Chain Conditions, Prime Ideals, and Classes of Locally Finite Lie Algebras

Abstract

A Noetherian (Artinian) Lie algebra satisfies the maximal (minimal) condition for ideals. Generalisations include quasi-Noetherian and quasi-Artinian Lie algebras. We study conditions on prime ideals relating these properties. We prove that the radical of any ideal of a quasi-Artinian Lie algebra is the intersection of finitely many prime ideals, and an ideally finite Lie algebra is quasi-Noetherian if and only if it is quasi-Artinian. Both properties are equivalent to soluble-by-finite. We also prove a structure theorem for serially finite Artinian Lie algebras.