{"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":54888,"journal":{"name":"Journal of Algebra and Its Applications","volume":"31 1","pages":""},"PeriodicalIF":0.5000,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Algebra and Its Applications","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1142/s0219498825502214","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
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.
期刊介绍：
The Journal of Algebra and Its Applications will publish papers both on theoretical and on applied aspects of Algebra. There is special interest in papers that point out innovative links between areas of Algebra and fields of application. As the field of Algebra continues to experience tremendous growth and diversification, we intend to provide the mathematical community with a central source for information on both the theoretical and the applied aspects of the discipline. While the journal will be primarily devoted to the publication of original research, extraordinary expository articles that encourage communication between algebraists and experts on areas of application as well as those presenting the state of the art on a given algebraic sub-discipline will be considered.
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