Closed ideals of Lie algebras
Let \(L\) be a Lie algebra and \(I\) be an ideal of \(L\). Define the closure of \(I\) in \(L\) as \(cl(I) = \{x \in L : \langle y \rangle^L \cap I \neq 0\) for all \(0 \neq y \in \langle x \rangle^L\}\), where \(\langle x \rangle^L\) is the ideal generated by \(x\). An ideal \(E\) of \(L\) is essential if the intersection of \(E\) with every nonzero ideal is nontrivial. Let \(cl^*(I) = \{x \in L : [x,E] \subseteq I\) for some essential ideal \(E\) of \(L\}\) be the *-closure of \(I\) in \(L\). \(I\) is said to be a closed ideal in \(L\) if and only if \(cl(I) = I\) and it is said to be *-closed in \(L\) if and only if \(cl^*(I) = I\). The authors show that \(cl(I) \subseteq cl^*(I)\) and that the closure of an ideal need not be an ideal, whereas the *-closure of an ideal is always an ideal. It is proved that for semisimple algebras, centralizer ideals, complement ideals, closed ideals and *-closed ideals are equivalent. It is also shown that *-closed ideals are always closed; closed ideals are always complement ideals and that the sum of two complement ideals need not be a complement ideal. The authors investigate chain conditions on closed and *-closed ideals and prove that for complement ideals the ascending chain condition and the descending chain condition are equivalent.
scientific article; zbMATH DE number 780124