{"title":"Fuzzy-Zorn引理及其应用","authors":"Xiao-long Xin, Yu-long Fu","doi":"10.1007/s11766-021-4043-8","DOIUrl":null,"url":null,"abstract":"<div><p>We introduced the fuzzy axioms of choice, fuzzy Zorn’s lemma and fuzzy well-ordering principle, which are the fuzzy versions of the axioms of choice, Zorn’s lemma and well-ordering principle, and discussed the relations among them. As an application of fuzzy Zorn’s lemma, we got the following results: (1) Every proper fuzzy ideal of a ring was contained in a maximal fuzzy ideal. (2) Every nonzero ring contained a fuzzy maximal ideal. (3) Introduced the notion of fuzzy nilpotent elements in a ring <i>R</i>, and proved that the intersection of all fuzzy prime ideals in a commutative ring <i>R</i> is the union of all fuzzy nilpotent elements in <i>R</i>. (4) Proposed the fuzzy version of Tychonoff Theorem and by use of fuzzy Zorn’s lemma, we proved the fuzzy Tychonoff Theorem.</p></div>","PeriodicalId":55568,"journal":{"name":"Applied Mathematics-A Journal of Chinese Universities Series B","volume":"36 4","pages":"521 - 536"},"PeriodicalIF":1.0000,"publicationDate":"2021-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11766-021-4043-8.pdf","citationCount":"3","resultStr":"{\"title\":\"Fuzzy Zorn’s lemma with applications\",\"authors\":\"Xiao-long Xin, Yu-long Fu\",\"doi\":\"10.1007/s11766-021-4043-8\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We introduced the fuzzy axioms of choice, fuzzy Zorn’s lemma and fuzzy well-ordering principle, which are the fuzzy versions of the axioms of choice, Zorn’s lemma and well-ordering principle, and discussed the relations among them. As an application of fuzzy Zorn’s lemma, we got the following results: (1) Every proper fuzzy ideal of a ring was contained in a maximal fuzzy ideal. (2) Every nonzero ring contained a fuzzy maximal ideal. (3) Introduced the notion of fuzzy nilpotent elements in a ring <i>R</i>, and proved that the intersection of all fuzzy prime ideals in a commutative ring <i>R</i> is the union of all fuzzy nilpotent elements in <i>R</i>. (4) Proposed the fuzzy version of Tychonoff Theorem and by use of fuzzy Zorn’s lemma, we proved the fuzzy Tychonoff Theorem.</p></div>\",\"PeriodicalId\":55568,\"journal\":{\"name\":\"Applied Mathematics-A Journal of Chinese Universities Series B\",\"volume\":\"36 4\",\"pages\":\"521 - 536\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2021-12-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://link.springer.com/content/pdf/10.1007/s11766-021-4043-8.pdf\",\"citationCount\":\"3\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Applied Mathematics-A Journal of Chinese Universities Series B\",\"FirstCategoryId\":\"1089\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s11766-021-4043-8\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Mathematics-A Journal of Chinese Universities Series B","FirstCategoryId":"1089","ListUrlMain":"https://link.springer.com/article/10.1007/s11766-021-4043-8","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
We introduced the fuzzy axioms of choice, fuzzy Zorn’s lemma and fuzzy well-ordering principle, which are the fuzzy versions of the axioms of choice, Zorn’s lemma and well-ordering principle, and discussed the relations among them. As an application of fuzzy Zorn’s lemma, we got the following results: (1) Every proper fuzzy ideal of a ring was contained in a maximal fuzzy ideal. (2) Every nonzero ring contained a fuzzy maximal ideal. (3) Introduced the notion of fuzzy nilpotent elements in a ring R, and proved that the intersection of all fuzzy prime ideals in a commutative ring R is the union of all fuzzy nilpotent elements in R. (4) Proposed the fuzzy version of Tychonoff Theorem and by use of fuzzy Zorn’s lemma, we proved the fuzzy Tychonoff Theorem.
期刊介绍：
Applied Mathematics promotes the integration of mathematics with other scientific disciplines, expanding its fields of study and promoting the development of relevant interdisciplinary subjects.
The journal mainly publishes original research papers that apply mathematical concepts, theories and methods to other subjects such as physics, chemistry, biology, information science, energy, environmental science, economics, and finance. In addition, it also reports the latest developments and trends in which mathematics interacts with other disciplines. Readers include professors and students, professionals in applied mathematics, and engineers at research institutes and in industry.
Applied Mathematics - A Journal of Chinese Universities has been an English-language quarterly since 1993. The English edition, abbreviated as Series B, has different contents than this Chinese edition, Series A.
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