Abstract :
In this paper the Lie algebras in which the lattice formed by the ideals is complemented or complemented and distributive are classified. Moreover, it is shown that the derived algebra (arbitrary characteristic) and the solvable radical (characteristic zero) can be characterized in terms of the ideal lattice structure. The relationship between Lie algebras having isomorphic lattices of ideals is also studied. It turns out that, over algebraically closed fields of characteristic zero, the Frattini ideal is preserved under ideal lattice isomorphisms and, as a consequence of this fact, the nilpotent radical is preserved by this kind of isomorphism when the codimension of the derived algebra is at least two.
