Existence and uniqueness of fuzzy ideals

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.