  In this paper, we introduce near perfect ideals and upper bounded ideals, and
study them as well as perfect ideals for finite dimensional Lie algebras. We
show that the largest perfect ideal and the largest near perfect ideal of a
finite dimensional Lie algebra always exist, and are equal to the smallest
ideal of the derived series, and the smallest ideal of the lower central
series, respectively. We call them the perfect radical and the near perfect
radical of that Lie algebra, respectively. A nonzero Lie algebra is solvable if
and only if its perfect radical is zero. The factor algebra of a Lie algebra by
its perfect radical is solvable. A nonzero Lie algebra is nilpotent if and only
if its near perfect radical is zero. The factor algebra of a Lie algebra by its
near perfect radical is nilpotent. We also show that the smallest upper bounded
ideal always exists, and is equal to the largest ideal of the upper central
series. For a nilpotent Lie algebra, there is only one upper bounded ideal,
i.e., the nilpotent Lie algebra itself.
