Fully noncentral Lie ideals and invariant additive subgroups in rings

Fuente: arXiv
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Hauptverfasser: Gardella, Eusebio, Lee, Tsiu-Kwen, Thiel, Hannes
Format: Preprint
Veröffentlicht: 2024
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author Gardella, Eusebio
Lee, Tsiu-Kwen
Thiel, Hannes
author_facet Gardella, Eusebio
Lee, Tsiu-Kwen
Thiel, Hannes
contents We prove conditions ensuring that a Lie ideal or an invariant additive subgroup in a ring contains all additive commutators. A crucial assumption is that the subgroup is fully noncentral, that is, its image in every quotient is noncentral. For a unital algebra over a field of characteristic $\neq 2$ where every additive commutator is a sum of square-zero elements, we show that a fully noncentral subspace is a Lie ideal if and only if it is invariant under all inner automorphisms. This applies in particular to zero-product balanced algebras.
format Preprint
id arxiv_https___arxiv_org_abs_2409_03362
institution arXiv
publishDate 2024
record_format arxiv
spellingShingle Fully noncentral Lie ideals and invariant additive subgroups in rings
Gardella, Eusebio
Lee, Tsiu-Kwen
Thiel, Hannes
Rings and Algebras
Operator Algebras
Primary 16N60, 16W10. Secondary 16S50, 16W20, 17B60
We prove conditions ensuring that a Lie ideal or an invariant additive subgroup in a ring contains all additive commutators. A crucial assumption is that the subgroup is fully noncentral, that is, its image in every quotient is noncentral. For a unital algebra over a field of characteristic $\neq 2$ where every additive commutator is a sum of square-zero elements, we show that a fully noncentral subspace is a Lie ideal if and only if it is invariant under all inner automorphisms. This applies in particular to zero-product balanced algebras.
title Fully noncentral Lie ideals and invariant additive subgroups in rings
topic Rings and Algebras
Operator Algebras
Primary 16N60, 16W10. Secondary 16S50, 16W20, 17B60
url https://arxiv.org/abs/2409.03362