{"title":"类似除法性质的研究","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":"{\"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}","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}
引用次数: 0
摘要
我们研究非交换环的弱可分性：如果对于所有非零的 a 和 x，存在 b、c，使得 x=ab+ca ，那么一个非琐环就是法德环。我们证明了非可分割环的性质，并构造了非可分割环的例子，以及非诺特环和非奥尔环的例子。
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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