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arXiv · 2409.03362

Fully noncentral Lie ideals and invariant additive subgroups in rings

Eusebio Gardella·Tsiu-Kwen Lee·Hannes Thiel
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Abstract

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.

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Eusebio Gardella, Tsiu-Kwen Lee, Hannes Thiel. 2024-09-05. Fully noncentral Lie ideals and invariant additive subgroups in rings. https://arxiv.org/abs/2409.03362

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