{"title":"关于对合除法环对称元素的一个注记","authors":"Vo Hoang Minh Thu","doi":"10.1007/s40306-021-00450-1","DOIUrl":null,"url":null,"abstract":"<div><p>Let <i>D</i> be a division ring with involution ⋆ and <i>S</i> the set of all symmetric elements of <i>D</i>. Assume that the center <i>F</i> of <i>D</i> is uncountable and <i>K</i> is a division subring of <i>D</i> containing <i>F</i>. The main aim of this note is to show that <i>S</i> is right algebraic over <i>K</i> if and only if so is <i>D</i>. This result allows us to construct an example of division rings <i>K</i> ⊂ <i>D</i> such that <i>D</i> is right algebraic but not left algebraic over <i>K</i>.</p></div>","PeriodicalId":45527,"journal":{"name":"Acta Mathematica Vietnamica","volume":null,"pages":null},"PeriodicalIF":0.3000,"publicationDate":"2021-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"A Note on Symmetric Elements of Division Rings with Involution\",\"authors\":\"Vo Hoang Minh Thu\",\"doi\":\"10.1007/s40306-021-00450-1\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Let <i>D</i> be a division ring with involution ⋆ and <i>S</i> the set of all symmetric elements of <i>D</i>. Assume that the center <i>F</i> of <i>D</i> is uncountable and <i>K</i> is a division subring of <i>D</i> containing <i>F</i>. The main aim of this note is to show that <i>S</i> is right algebraic over <i>K</i> if and only if so is <i>D</i>. This result allows us to construct an example of division rings <i>K</i> ⊂ <i>D</i> such that <i>D</i> is right algebraic but not left algebraic over <i>K</i>.</p></div>\",\"PeriodicalId\":45527,\"journal\":{\"name\":\"Acta Mathematica Vietnamica\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.3000,\"publicationDate\":\"2021-09-02\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Acta Mathematica Vietnamica\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s40306-021-00450-1\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Mathematica Vietnamica","FirstCategoryId":"1085","ListUrlMain":"https://link.springer.com/article/10.1007/s40306-021-00450-1","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
A Note on Symmetric Elements of Division Rings with Involution
Let D be a division ring with involution ⋆ and S the set of all symmetric elements of D. Assume that the center F of D is uncountable and K is a division subring of D containing F. The main aim of this note is to show that S is right algebraic over K if and only if so is D. This result allows us to construct an example of division rings K ⊂ D such that D is right algebraic but not left algebraic over K.
期刊介绍:
Acta Mathematica Vietnamica is a peer-reviewed mathematical journal. The journal publishes original papers of high quality in all branches of Mathematics with strong focus on Algebraic Geometry and Commutative Algebra, Algebraic Topology, Complex Analysis, Dynamical Systems, Optimization and Partial Differential Equations.