Ascending chain conditions on special classes of ideals of Lie algebras (Q1199180)

The authors consider the classes Max-CI, Max-ESS, and Max-SMI of Lie algebras with the ascending chain condition on complement ideals, essential ideals, and small ideals, respectively. Let \(L\) be a Lie algebra. Suppose that every quotient \(L/I\) by nonzero ideals \(I\) (of some special type) satisfies some ascending chain condition. The authors prove that \(L\) itself satisfies the chain condition for Max-CI (\(I\) a complement ideal) and Max-SMI (\(I\) a small ideal). They show a negative result for the class Max-c of Lie algebras with the ascending chain condition on centralizer ideals, and give an answer to Question 3 of their previous paper [Hiroshima Math. J. 19, 397-407 (1989; Zbl 0697.17010)]. A Lie algebra analogue is shown for the result of \textit{V. P. Camillo} [Glasg. Math. J. 16, 32-33 (1975; Zbl 0315.13013)].