{"title":"Study of a division-like property","authors":"Robin Khanfir, Béranger Seguin","doi":"10.1142/s0219498825502214","DOIUrl":null,"url":null,"abstract":"<p>We study a weak divisibility property for noncommutative rings: A nontrivial ring is <i>fadelian</i> if for all nonzero <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>a</mi></math></span><span></span> and <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>x</mi></math></span><span></span> there exist <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>b</mi><mo>,</mo><mi>c</mi></math></span><span></span> such that <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>x</mi><mo>=</mo><mi>a</mi><mi>b</mi><mo stretchy=\"false\">+</mo><mi>c</mi><mi>a</mi></math></span><span></span>. We prove properties of fadelian rings and construct examples thereof which are not division rings, as well as non-Noetherian and non-Ore examples.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1142/s0219498825502214","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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Abstract
We study a weak divisibility property for noncommutative rings: A nontrivial ring is fadelian if for all nonzero and there exist such that . We prove properties of fadelian rings and construct examples thereof which are not division rings, as well as non-Noetherian and non-Ore examples.
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