Abstract
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.
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Supported by the National Natural Science Foundation of China(11971384), and by the grant of Natural Science Basic Research Program of Shaanxi (Program No. 2021JM-137), the Fundamental Research Funds for the Central Universities under grant QTZX2106, China 111 Project (B16037) and OPPO Research Fund.
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Xin, Xl., Fu, Yl. Fuzzy Zorn’s lemma with applications. Appl. Math. J. Chin. Univ. 36, 521–536 (2021). https://doi.org/10.1007/s11766-021-4043-8
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DOI: https://doi.org/10.1007/s11766-021-4043-8