{"title":"Fully noncentral Lie ideals and invariant additive subgroups in rings","authors":"Eusebio Gardella, Tsiu-Kwen Lee, Hannes Thiel","doi":"arxiv-2409.03362","DOIUrl":null,"url":null,"abstract":"We prove conditions ensuring that a Lie ideal or an invariant additive\nsubgroup in a ring contains all additive commutators. A crucial assumption is\nthat the subgroup is fully noncentral, that is, its image in every quotient is\nnoncentral. For a unital algebra over a field of characteristic $\\neq 2$ where every\nadditive commutator is a sum of square-zero elements, we show that a fully\nnoncentral subspace is a Lie ideal if and only if it is invariant under all\ninner automorphisms. This applies in particular to zero-product balanced\nalgebras.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"53 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Rings and Algebras","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.03362","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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.