Intersections of subideals of Lie algebras (Q912199)

Let \({\mathfrak L}_{\infty}\), \({\mathfrak L}_{\infty}(asc)\), \({\mathfrak D}(asc,si)\) and \({\mathfrak M}\) denote the class of Lie algebras in which the join of any collection of subideals is always a subideal, the intersection of any collection of ascendant subalgebras is always ascendant, every ascendant subalgebra is a subideal and the steps of all subideals have an upper bound, respectively. It is first proved that in any Lie algebra the intersection of any collection of descendant (weakly descendant, serial, weakly serial) subalgebras is always descendant (weakly descendant, serial, weakly serial). \({\mathfrak L}_{\infty}\) is characterized as the class of Lie algebras in which every descendant subalgebra is a subideal. It is shown that if L is a Lie algebra with an ideal I having a composition series of finite length and L/I\(\in {\mathfrak L}_{\infty}\) (\({\mathfrak L}_{\infty}(asc)\), \({\mathfrak M}\), \({\mathfrak D}(asc,si)\), then \(L\in {\mathfrak L}_{\infty}\) (\({\mathfrak L}_{\infty}(asc)\), \({\mathfrak M}\), \({\mathfrak D}(asc,si))\). It is proved that if \(L\in {\mathfrak L}_{\infty}\) has an abelian ideal of codimension 1, then \(L\in {\mathfrak M}\). Finally, a sufficient condition is given for Lie algebras in \({\mathfrak L}_{\infty}\) to be nilpotent.