The Lie Inner Ideal Structure of Associative Rings Revisited

Benkart, Georgia; López, Antonio Fernández

doi:10.1080/00927870802545729

Cited by 21 publications

(18 citation statements)

References 17 publications

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“…Note that ad a is nonzero and has finite rank for every nonzero a ∈ L. However, L is infinite-dimensional. [5,Theorem 6.6], we observe that the only possibility for the nonexistence of a nontrivial abelian inner ideal of finite dimension is that L is the finitary orthogonal algebra defined by an infinite-dimensional vector space X with a nondegenerate symmetric bilinear form not containing a totally isotropic subspace of dimension at least 2. But the existence of a 2-dimensional totally isotropic subspace is equivalent to the existence of a skew operator a of rank 2 such that a 2 = 0.…”

Section: Results On Lie Algebrasmentioning

confidence: 99%

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On a class of finitary Lie algebras characterized through derivations

Brešar

López

2010

Proc. Amer. Math. Soc.

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Abstract. Let L be an infinite-dimensional simple Lie algebra over a field of characteristic 0. Then there exist a derivation d on L and n ≥ 2 such that d n is a nonzero finite rank map if and only if L is finitary and contains a nonzero finite-dimensional abelian inner ideal. This is a partial statement of our main theorem. As auxiliary results needed for the proof we establish some properties of derivations in general nonassociative algebras.

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Section: Results On Lie Algebrasmentioning

confidence: 99%

“…The first topic concerns finitary Lie algebras (see, e.g., [1]) and Lie algebras with minimal inner ideals, a notion introduced by G. Benkart in [4] (see, e.g., [5,8]). The second topic concerns derivations of algebras such that some of their powers are nonzero finite rank operators (see, e.g., [6,7]).…”

Section: Introductionmentioning

confidence: 99%

On a class of finitary Lie algebras characterized through derivations

Brešar

López

2010

Proc. Amer. Math. Soc.

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“…We finish with an application to the Lie inner ideal structure of simple rings. Inner ideals were first systematically studied by Benkart [2], see also [3,4] for some recent development. (iii) ⇒ (i).…”

Section: 5mentioning

confidence: 99%

Maximal zero product subrings and inner ideals of simple rings

Baranov

López

2019

Journal of Algebra

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Let Q be a (non-unital) simple ring. A nonempty subset S of Q is said to have zero product if S 2 = 0. We classify all maximal zero product subsets of Q by proving that the map R → R ∩ LeftAnn(R) is a bijection from the set of all proper nonzero annihilator right ideals of Q onto the set of all maximal zero product subsets of Q. We also describe the relationship between the maximal zero product subsets of Q and the maximal inner ideals of its associated Lie algebra.

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“…It was shown in [13] that inner ideals play role similar to that of one-sided ideals in associative algebras and can be used to develop Artinian structure theory for Lie algebras. Inner ideals of classical Lie algebras were classified by Benkart and Fernndez Lpez [7,9], using the fact that these algebras can be obtained as the derived Lie subalgebras of (involution) simple Artinian associative rings. In this paper we use a similar approach to study inner ideals of the derived Lie subalgebras of finite dimensional associative algebras.…”

Section: Introductionmentioning

confidence: 99%