{"title":"New characterization of $(b,c)$-inverses through polarity","authors":"Btissam Laghmam, Hassane Zguitti","doi":"arxiv-2409.11987","DOIUrl":null,"url":null,"abstract":"In this paper we introduce the notion of $(b,c)$-polar elements in an\nassociative ring $R$. Necessary and sufficient conditions of an element $a\\in\nR$ to be $(b,c)$-polar are investigated. We show that an element $a\\in R$ is\n$(b,c)$-polar if and only if $a$ is $(b,c)$-invertible. In particular the\n$(b,c)$-polarity is a generalization of the polarity along an element\nintroduced by Song, Zhu and Mosi\\'c [14] if $b=c$, and the polarity introduced\nby Koliha and Patricio [10]. Further characterizations are obtained in the\nBanach space context.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"14 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-18","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-2409.11987","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
In this paper we introduce the notion of $(b,c)$-polar elements in an
associative ring $R$. Necessary and sufficient conditions of an element $a\in
R$ to be $(b,c)$-polar are investigated. We show that an element $a\in R$ is
$(b,c)$-polar if and only if $a$ is $(b,c)$-invertible. In particular the
$(b,c)$-polarity is a generalization of the polarity along an element
introduced by Song, Zhu and Mosi\'c [14] if $b=c$, and the polarity introduced
by Koliha and Patricio [10]. Further characterizations are obtained in the
Banach space context.