Jacobson’s refinement of Engel’s theorem for Leibniz algebras

We develop Jacobson’s refinement of Engel’s Theorem for Leibniz algebras. We then note some consequences of the result. Since Leibniz algebras were introduced in [Loday 1993] as a noncommutative generalization of Lie algebras, one theme has been to extend Lie algebra results to Leibniz algebras. In particular, Engel’s theorem has been extended in [Ayupov and Omirov 1998; Barnes 2011; Patsourakos 2007]. In the second of these works, the classical Engel’s theorem is used to give a short proof of the result for Leibniz algebras. The proofs in the other two papers do not use the classical theorem and, therefore, the Lie algebra result is included in the result. In this note, we give two proofs of the generalization to Leibniz algebras of Jacobson’s refinement to Engel’s theorem, a short proof which uses Jacobson’s theorem and a second proof which does not use it. It is interesting to note that the technique of reducing the problem to the special Lie algebra case significantly shortens the proof for the general Leibniz algebras case. This approach has been used in a number of situations [Barnes 2011]. We also note some standard consequences of this theorem. The proofs of the corollaries are exactly as in Lie algebras (see [Kaplansky 1971]). Our result can be used to directly show that the sum of nilpotent ideals is nilpotent, and hence one has a nilpotent radical. In this paper, we consider only finite dimensional algebras and modules over a field F. An algebra A is called Leibniz if it satisfies x.yz/D.x y/zC y.x z/. Denote by Ra and L a, respectively, right and left multiplication by a2 A. Then RbcD Rc RbC L b Rc;