In this note we investigate Lie
algebras which satisfy the descending chain condition on ideals of ideals. We show
that a Lie algebra L satisfies this descending chain condition if and only if the
following two conditions hold: (i) L contains a finite dimensional solvable ideal N
such that every solvable ideal of L is contained in N, and (ii) L∕N is a subdirect sum
of a finite number of prime algebras satisfying the descending chain condition. We
also show that if L is a prime algebra with this chain condition then there exists a Lie
algebra B, which is either simple or the tensor product of a simple Lie algebra with a
truncated polynomial algebra, such that L is isomorphic to a subalgebra of Der B
containing adB.
