On the Definition of Atanassov’s Intuitionistic Fuzzy Subrings and Ideals

On the basis of the concept of grades of a fuzzy point to belongingness (element of) or quasi-coincident (q) or belongingness and quasi-coincident (element of ˄q) or belongingness or quasi-coincident (element of ˅q) in an intuitionistic fuzzy set of a ring, the notion of a (α,β)-intuitionistic fuzzy subring and ideal is introduced by applying the Lukasiewicz 3-valued implication operator. Using the notion of fuzzy cut set of an intuitionistic fuzzy set, the support and α-level set of an intuitionistic fuzzy set are defined and it is established that, for a α≠element of ˄q, the support of a (α,β)-intuitionistic fuzzy ideal of a ring is an ideal of the ring. It is also established that the level sets of an intuitionistic fuzzy ideal with thresholds (s, t) of a ring is an ideal of the ring. We investigate that an intuitionistic fuzzy set A of a ring is a (element of, element of) (or (element of, element of ˅q ) or (element of ˄q,element of)-intuitionistic fuzzy ideal of the ring if and only if A is an intuitionistic fuzzy ideal with thresholds (0,1) (or (0,0:5) or (0:5,1)) of the ring respectively. We also establish that A is a (element of, element of) (or (element of, element of ˅q ) or (element of ˄q,element of) )-intuitionistic fuzzy ideal of the ring if and only if for any a element of (0,1] (or a element of (0,0.5] or a element of (0.5,1] ), Aa is a fuzzy ideal of the ring. Finally, we investigate that an intuitionistic fuzzy set of a ring is an intuitionistic fuzzy ideal with thresholds (s, t) of the ring if and only if for any a element of (s,t], the cut set Aa is a fuzzy ideal of R.