On Lie algebras with finiteness conditions (Q788816)

In this paper we consider three types of questions concerning the ideal and subideal structure of Lie algebras with certain finiteness conditions. First we consider the question of finding conditions under which the join of subideals of a Lie algebra is a subideal. Theorem 1.2 shows that when L is a Lie algebra over a field of characteristic zero and \(\{H_{\lambda}| \lambda \in \Lambda \}\) is a set of subideals of L with J their join, J is a subideal of L if and only if the set of subideals of L lying in J has a maximal element. We also find another condition under which the join of subideals is a subideal by imposing conditions on the circle product \(H{\mathbb{O}}K=[H,K]^{<H,K>}\) of subideals. The second problem is to investigate the structure of Lie algebras with a certain chain condition on subideals using the notion of prime ideals and prime algebras (defined by analogy with associative rings). In particular Theorem 2.1 proves that when L is a Lie algebra over any field and \({\mathfrak X}\) is one of \(\max -\triangleleft^ n(n\geq 2)\), max-si, \(\min - \triangleleft^ n\), min-si, \(L\in {\mathfrak X}\) if and only if (i) \(\sigma\) (L) is a finite-dimensional soluble ideal of L. (ii) L/\(\sigma\) (L) is a subdirect sum of a finite number of prime algebras in \({\mathfrak X}\). \(\sigma\) (L) denotes a generalization of soluble radical. Thirdly, we generalize the minimal condition on ideals, leading to a new class of quasi-Artinian algebras which possesses several of the main properties of \(\min -\triangleleft\). Theorems 3.2, 3.3 prove that the class of quasi-Artinian algebras is Q-closed and that a locally nilpotent quasi-Artinian Lie algebra is soluble. Finally we remark that it is possible to define the notion of quasi-Artinian groups in an analogous way and the proofs of Theorems 3.2, 3.3 carry over in this case without difficulties.