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 (∈) or quasi-coincident (q) or belongingness and quasi-coincident (∈ ∧q) or belongingness or quasi-coincident (∈ ∨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 α 6=∈ ∧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 (∈,∈) (or (∈,∈ ∨q ) or (∈ ∧q,∈) )-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 (∈,∈) (or (∈,∈ ∨q ) or (∈ ∧q,∈) )-intuitionistic fuzzy ideal of the ring if and only if for any a ∈ (0,1] (or a ∈ (0,0.5] or a ∈ (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 ∈ (s, t], the cut set Aa is a fuzzy ideal of R. 2010 Mathematics Subject Classification: 08A72, 16D25.