Classes of generalized soluble Lie algebras (Q1079646)

The author introduces and investigates various classes of Lie algebras generalizing the class of residually central Lie algebras. Most of these are classes of generalized soluble Lie algebras. He proves that if \({\mathfrak X}\) is a class of Lie algebras having abelian series (resp. abelian ideal series or central series), then \({\mathfrak X}\) is L-closed. He also proves that if L has a central (resp. descending central) series, then every finite-dimensional subalgebra of L is serial (resp. descendant). Furthermore he shows that \[ {\mathfrak R}_{(*)}\cap {\mathfrak M}^{(*)}=\grave EA\text{ and }\acute E(\triangleleft)A\cap Min- \triangleleft ={\mathfrak R}^{(*)}_{(*)}\cap Min-\triangleleft, \] where the classes \(\grave EA\), \(\acute E(\triangleleft)A\), \({\mathfrak R}_{(*)}\), \({\mathfrak R}^{(*)}_{(*)}\) and \({\mathfrak M}^{(*)}\) are defined as follows: \(L\in \grave EA\) if L has a descending series with abelian factors. \(L\in \acute E(\triangleleft)A\) if L has an ascending series of ideals with abelian factors. \(L\in {\mathfrak R}_{(*)}\) if \(x\in L\setminus \{0\}\) implies \(x\not\in [x, L^{(*)}]^ L,\) \(L\in {\mathfrak R}^{(*)}_{(*)}\) if \(x\in L\setminus \{0\}\) implies \(x\not\in ([x, L^{(*)}]^ L)^{(*)},\) \(L\in {\mathfrak M}^{(*)}\) if for any descending chain \(I_ 1\supseteq I_ 2\supseteq..\). of ideals of L contained in \(L^{(*)}\) there exists an integer \(n=n(I_ 1,I_ 2,...)>0\) such that \(I_ n/\cap_{i\geq 1}I_ i\leq \zeta_ 1(L^{(*)})/\cap_{i\geq 1}I_ i\), where \(L^{(*)}\) denotes the intersection of all the terms of the transfinite derived series of L.