{"title":"Group invertibility of the sum in rings and its applications","authors":"Huanyin Chen, Dayong Liu, Marjan Sheibani","doi":"10.1515/gmj-2024-2010","DOIUrl":null,"url":null,"abstract":"We present necessary and sufficient conditions under which the sum of two group invertible elements in a ring is group invertible. As applications, we establish the existence of group inverses of certain <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mn>2</m:mn> <m:mo>×</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2010_eq_0193.png\" /> <jats:tex-math>{2\\times 2}</jats:tex-math> </jats:alternatives> </jats:inline-formula> block-operator matrices over a Banach space. These generalize the known results, e.g., Zhou, Chen and Zhu (<jats:italic>Comm. Algebra</jats:italic> 48 (2020), 676–690) and Benítez, Liu and Zhu (<jats:italic>Linear Multilinear Algebra</jats:italic> 59 (2011), 279–289).","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/gmj-2024-2010","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 present necessary and sufficient conditions under which the sum of two group invertible elements in a ring is group invertible. As applications, we establish the existence of group inverses of certain 2×2{2\times 2} block-operator matrices over a Banach space. These generalize the known results, e.g., Zhou, Chen and Zhu (Comm. Algebra 48 (2020), 676–690) and Benítez, Liu and Zhu (Linear Multilinear Algebra 59 (2011), 279–289).
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