Rough Intuitionistic Fuzzy Ideals in a Ring

he theory of fuzzy sets was ntroduced by Zadeh [22] in 1965 as an attempt to study the vagueness and uncertainity in real world problems. A.Rosenfeld [16] applied the notion of fuzzy sets and introduced the notion of fuzzy subgroups in groups. Since then various classical algebraic systems have been fuzzified. The theory of rough sets introduced by Z.Pawlak [14] in 1982 is another independent method to deal the vagueness and uncertainty. Z.Pawlak used equivalence classes in a set as the building blocks for the construction of lower and upper approximations of a set. R.Biswas and S.Nanda [5] introduced the concept of lower and upper approximation of a subgroup of a group. N.Kuroki [12] gave the notion of rough ideal in a semigroup. The concept of rough group was also considered by N.Kuroki and J.N.Moderson [13]. B.Davvaz [6] has introduced roughness in rings. O.Kazanci and B.Davvaz [10] introduced the notion of rough prime ideals and rough primary ideals in rings. The authors V.Selvan and G.Senthil Kumar [17, 18] have studied the lower and upper approximations of ideals and fuzzy ideals in a semiring. As a generalization of fuzzy sets, the concept of intuitionistic fuzzy set was introduced by K.T.Atanassov in [1]. R.Biswas [4] used the concept of intuitionistic fuzzy set to the theory of groups and studied the intuitionistic fuzzy subgroups of a group. The concept of intuitionistic fuzzy R-subgroup of a near ring was given by Y.H.Yon, Y.B.Jun and K.H.Kim [21]. The concept of   ,   -cut of intuitionistic fuzzy ideals in a ring was given by D.K.Basnet [3]. The notion of rough intuitionistic fuzzy ideal in a semi group was given by J.Gosh and T.K.Samanta [9]. In this paper, in section 3 we prove that any intuitionistic fuzzy subring (ideal) of a ring is an upper (lower) approximation intuitionistic fuzzy subring (ideal) of the ring. In section 4 of this paper, we introduce the rough intuitionistic fuzzy prime ideal of a ring. Also we discuss the relationship between upper and lower intuitionistic fuzzy ideals (prime ideals) and the upper and lower approximation of their homomorphic images.