Essentially finitely generated Lie algebras (Q1355503)

An ideal \(I\) of a Lie algebra \(L\) is called essential if \(I\) intersects every nonzero ideal of \(L\) nontrivially, and \(L\) is called essentially finitely generated if \(L\) has a finitely generated essential ideal. An \(L\)-module \(M\) is essentially finitely generated over \(L\) if \(M\) has a finitely generated essential \(L\)-submodule. The author shows the following results: (a) If an ideal \(I\) of \(L\) and \(L/I\) are essentially finitely generated, then so is \(L\). (b) Let \(L\) be a semisimple Lie algebra. Then \(L\) satisfies the minimal condition on centralizer ideals if and only if every prime ideal is essentially finitely generated over \(L\). (c) A Lie algebra \(L\) is essentially finitely generated if the intersection of finitely many prime ideals is zero. The result (c) gives an affirmative answer to the Question 4 raised by \textit{F. A. M. Aldosray} and \textit{I. Stewart} [Hiroshima Math. J. 19, No. 2, 397-407 (1989; Zbl 0697.17010)].