{"title":"A Characterisation of the Quaternions Using Commutators","authors":"E. Kleinfeld, Yoav Segev","doi":"10.1353/mpr.2022.0000","DOIUrl":null,"url":null,"abstract":"<p>Abstract:</p><p>Let <i>R</i> be an associative ring with 1, which is not commutative. Assume that any non-zero commutator <i>v</i> ∈ <i>R</i> satisfies: <i>v</i>2 is in the centre of <i>R</i>, and <i>v</i> is not a zero divisor.</p><p>We prove that <i>R</i> has no zero divisors, and that if char(<i>R</i>) ≠ 2, then the localisation of <i>R</i> at its centre is a quaternion division algebra. Our proof is elementary and self contained.</p>","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"99 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2021-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Proceedings of the Royal Irish Academy","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1353/mpr.2022.0000","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 1
Abstract
Abstract:
Let R be an associative ring with 1, which is not commutative. Assume that any non-zero commutator v ∈ R satisfies: v2 is in the centre of R, and v is not a zero divisor.
We prove that R has no zero divisors, and that if char(R) ≠ 2, then the localisation of R at its centre is a quaternion division algebra. Our proof is elementary and self contained.
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