Quasi-Noetherian Lie algebras: Correction and further results

We correct [3, Proposition 3.2] by showing that the class of quasi-Noetherian Lie algebras is not E-closed. We repair the resulting gap in Example 5.1 of that paper by proving that any simple-by-soluble Lie algebra is quasi-Noetherian. More generally, any Noetherian-by-quasi-Noetherian Lie algebra is quasi-Noetherian. We also prove that any quasi-Noetherian-by-soluble Lie algebra is quasi-Noetherian, and prove E-closure for analogues of the quasi-Noetherian property for modules over associative rings and modules over Lie algebras. Using a wreath product for Lie algebras we prove that the quasi-Noetherian property is not inherited by ideals. Indeed, the derived algebra of a quasi-Noetherian Lie algebra need not be quasi-Noetherian. An analogous example is constructed for quasi-Artinian Lie algebras, which also shows that the Artinian and quasi-Artinian properties are not inherited by ideals of finite codimension.