Vol. 73, No. 1, 1977

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Lie algebras with descending chain condition

J. Marshall Osborn

Vol. 73 (1977), No. 1, 155–159

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.

Mathematical Subject Classification 2000
Primary: 17B05
Received: 20 July 1976
Published: 1 November 1977
J. Marshall Osborn