{"title":"论交换环的准最大图式","authors":"Murat Alan, Mesut Kılıç, Suat Koç, Ünsal Tekir","doi":"10.7546/crabs.2023.12.01","DOIUrl":null,"url":null,"abstract":"Let $$R$$ be a commutative ring with $$1\\ne 0$$. A proper ideal $$I$$ of $$R$$ is said to be a quasi maximal ideal if for every $$a\\in R-I$$, either $$I+Ra=R$$ or $$I+Ra$$ is a maximal ideal of $$R$$. This class of ideals lies between 2-absorbing ideals and maximal ideals which is different from prime ideals. In addition to give fundamental properties of quasi maximal ideals, we characterize principal ideal UN-rings with $$\\sqrt{0}^2=(0)$$, direct product of two fields, and Noetherian zero dimensional modules in terms of quasi maximal ideals.","PeriodicalId":104760,"journal":{"name":"Proceedings of the Bulgarian Academy of Sciences","volume":"23 2","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2023-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On Quasi Maximal Ideals of Commutative Rings\",\"authors\":\"Murat Alan, Mesut Kılıç, Suat Koç, Ünsal Tekir\",\"doi\":\"10.7546/crabs.2023.12.01\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let $$R$$ be a commutative ring with $$1\\\\ne 0$$. A proper ideal $$I$$ of $$R$$ is said to be a quasi maximal ideal if for every $$a\\\\in R-I$$, either $$I+Ra=R$$ or $$I+Ra$$ is a maximal ideal of $$R$$. This class of ideals lies between 2-absorbing ideals and maximal ideals which is different from prime ideals. In addition to give fundamental properties of quasi maximal ideals, we characterize principal ideal UN-rings with $$\\\\sqrt{0}^2=(0)$$, direct product of two fields, and Noetherian zero dimensional modules in terms of quasi maximal ideals.\",\"PeriodicalId\":104760,\"journal\":{\"name\":\"Proceedings of the Bulgarian Academy of Sciences\",\"volume\":\"23 2\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2023-12-21\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the Bulgarian Academy of Sciences\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.7546/crabs.2023.12.01\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the Bulgarian Academy of Sciences","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.7546/crabs.2023.12.01","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Let $$R$$ be a commutative ring with $$1\ne 0$$. A proper ideal $$I$$ of $$R$$ is said to be a quasi maximal ideal if for every $$a\in R-I$$, either $$I+Ra=R$$ or $$I+Ra$$ is a maximal ideal of $$R$$. This class of ideals lies between 2-absorbing ideals and maximal ideals which is different from prime ideals. In addition to give fundamental properties of quasi maximal ideals, we characterize principal ideal UN-rings with $$\sqrt{0}^2=(0)$$, direct product of two fields, and Noetherian zero dimensional modules in terms of quasi maximal ideals.
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