×

Fuzzy ideals and fuzzy prime ideals of a ring. (English) Zbl 0772.16002

Firstly, the authors focus on the generalization of the well-known classical property: the union of two ideals of a ring is again an ideal iff one of them is contained in the other. By means of a counterexample it is proven that this property does not hold in general for fuzzy ideals. Some sufficient conditions to keep the property in the fuzzy case are outlined. Furthermore the authors realize some fuzzifications of results in classical ring theory about fuzzy prime ideals and fuzzy maximal ideals, among them a fuzzified version of the theorem of Cohen about the characterization of Noetherian rings.
Reviewer: E.Kerre (Gent)

MSC:

16D25 Ideals in associative algebras
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Anthony, J. M.; Sherwood, H., A characterization of fuzzy subgroups, Fuzzy Sets and Systems, 7, 297-305 (1982) · Zbl 0524.20050
[2] Burton, D. M., A First Course in Rings and Ideals (1970), Addison-Wesley: Addison-Wesley Reading, MA · Zbl 0204.05601
[3] Das, P. S., Fuzzy groups and level subgroups, J. Math. Anal. Appl., 84, 264-269 (1981) · Zbl 0476.20002
[4] Dixit, V. N.; Kumar, R.; Ajmal, N., Level subgroups and union of fuzzy subgroups, Fuzzy Sets and Systems, 37, 359-371 (1990) · Zbl 0728.20061
[5] V.N. Dixit, R. Kumar and N. Ajmal, On fuzzy rings, Fuzzy Sets and Systems; V.N. Dixit, R. Kumar and N. Ajmal, On fuzzy rings, Fuzzy Sets and Systems · Zbl 0771.13012
[6] Kuroki, N., On fuzzy ideals and fuzzy bi-ideals in semigroups, Fuzzy Sets and Systems, 5, 203-215 (1981) · Zbl 0452.20060
[7] Kuroki, N., Fuzzy semiprime ideals in semigroups, Fuzzy Sets and Systems, 8, 71-79 (1982) · Zbl 0488.20049
[8] Liu, Wang-jin, Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Sets and Systems, 8, 133-139 (1982) · Zbl 0488.20005
[9] Mukherjee, N. P.; Bhattacharya, P., Fuzzy normal subgroups and fuzzy cosets, Inform. Sci., 34, 225-239 (1984) · Zbl 0568.20002
[10] Mukherjee, T. K.; Sen, M. K., On fuzzy ideals of a ring (1), Fuzzy Sets and Systems, 21, 99-104 (1987) · Zbl 0617.13001
[11] Rosenfeld, A., Fuzzy groups, J. Math. Anal. Appl., 35, 512-517 (1971) · Zbl 0194.05501
[12] Swamy, K. L.N.; Swamy, U. M., Fuzzy prime ideals of rings, J. Math. Anal. Appl., 134, 94-103 (1988) · Zbl 0663.16019
[13] Yue, Zhang, Prime L-fuzzy ideals and primary L-fuzzy ideals, Fuzzy Sets and Systems, 27, 345-850 (1988) · Zbl 0663.13001
[14] Zadeh, L., Fuzzy sets, Inform. and Control, 8, 338-353 (1965) · Zbl 0139.24606
[15] Zimmermann, H.-J., Fuzzy Set Theory - and its Applications (1985), Kluwer/Nijhoff: Kluwer/Nijhoff Dordrecht · Zbl 0578.90095
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.