Existence and uniqueness of fuzzy ideals

doi:10.1016/j.fss.2004.04.009

Fuzzy Sets and Systems

Volume 149, Issue 3, 1 February 2005, Pages 527-541

https://doi.org/10.1016/j.fss.2004.04.009 Get rights and content

Abstract

Let R be a commutative ring. The idea of identifying a fuzzy ideal of R with its corresponding chain of level ideals and its range of values, although widely used in the literature for the study of fuzzy ideals, has never been investigated in a general setting before. We investigate in this paper the more general problem of characterization of all those fuzzy ideals that can be identified with a given arbitrary family C of subsets of R together with a given arbitrary subset S of [0,1]. Necessary and sufficient conditions for the existence and uniqueness of such fuzzy ideals are established. In particular, we obtain that uniqueness always holds true for Artinian rings. Moreover, examples are presented to further support our results and shed more light on the above-mentioned problem of identification. We finish by raising a natural question that is left as an open problem for further investigation.

References (26)

P.S. Das
Fuzzy groups and level subgroups
J. Math. Anal. Appl
(1981)
V.N. Dixit et al.
On fuzzy rings
Fuzzy Sets and Systems
(1992)
R. Kumar
Certain fuzzy ideals of rings redefined
Fuzzy Sets and Systems
(1992)
W.J. Liu
Invariant subgroups and fuzzy ideals
Fuzzy Sets and Systems
(1982)
T.K. Mukherjee et al.
On fuzzy ideals of a ring (1)
Fuzzy Sets and Systems
(1987)
T.K. Mukherjee et al.
Rings with chain conditions
Fuzzy Sets and Systems
(1991)
D.A. Ralescu
A generalization of the representation theorem
Fuzzy Sets and Systems
(1992)
A. Rosenfeld
Fuzzy groups
J. Math. Anal. Appl
(1971)
B. Šešelja et al.
On a representation of posets by fuzzy sets
Fuzzy Sets and Systems
(1998)
B. Šešelja et al.
Completion of ordered structures by cuts of fuzzy setsan overview
Fuzzy Sets and Systems
(2003)

B. Šešelja et al.

Representing ordered structures by fuzzy setsan overview

Fuzzy Sets and Systems

(2003)

L.A. Zadeh

Fuzzy sets

Inform. Control

(1965)

D. Cox et al.

Ideals, varieties, and algorithms, An introduction to computational algebraic geometry and commutative algebra, Undergraduate Texts in Mathematics

(1992)

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Existence and uniqueness of fuzzy ideals

Abstract

J. Math. Anal. Appl

Fuzzy Sets and Systems

Fuzzy Sets and Systems

Fuzzy Sets and Systems

Fuzzy Sets and Systems

Fuzzy Sets and Systems

Fuzzy Sets and Systems

J. Math. Anal. Appl

Fuzzy Sets and Systems

Fuzzy Sets and Systems

Fuzzy Sets and Systems

Inform. Control

Ideals, varieties, and algorithms, An introduction to computational algebraic geometry and commutative algebra, Undergraduate Texts in Mathematics