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

Three Minimax Ideal Relations of Lie Algebras