International Mathematical Forum, 4, 2009, no. 13, 623 - 630 Interval Valued (∈, ∈∨ q)-Fuzzy Ideal in Rings Dong Soo Lee and Chul Hwan Park Department of Mathematics University of Ulsan, Ulsan 680-749, Korea dslee@ulsanl.ac.kr skyrosemary@gmail.com Abstract The notion of an interval-valued (∈, ∈∨q)-fuzzy subring(ideal,prime) in ring is introduced and their characterizations are investigated. Mathematics Subject Classification: 03E72,16D25 Keywords: quasi-coincidence,interval-valued (∈, ∈∨q)-fuzzy subring(ideal, prime) 1 Introduction Fuzzy set was initiated by Zadeh[10] and so many researchers were conducted on the generalizations of the notion of fuzzy sets. In [11], Zadeh made an extension of the concept of a fuzzy set by an interval-valued fuzzy set. Liu ap- plied the concept of fuzzy sets to the theory of rings and introduced the notions of fuzzy subring and fuzzy ideal of a ring[8]. This concept discussed futher by many researchers[1, 3, 4, 6, 7]. In [9], Ming and Ming introduced introduced the cocept of quasi-coincidence of a fuzzy point with a fuzzy suset. Based on quasi-coincidence, S.K. Bhakat and P.Das[2] introduced a new type fuzzy subring(ideal,prime) of ring called an (∈, ∈∨q)-fuzzy subring(ideal,prime). In this paper, we concenterate on the quasi-coincidence of a fuzzy interval value with an interval valued fuzzy set and introduced the notions of (∈, ∈ ∨q)-fuzzy subring(ideal,prime) which is an extende notion of (∈, ∈∨q)-fuzzy subring(ideal,prime). And we give some interesting properties are investigated. 2 Prelminaries Let R be a ring. By a subring of R we mean a nonempty subset S of R such that S is closed under the operations of addition and multiplication in R. A subring I of a ring R is called an ideal of R if for all x ∈ R, r ∈ I, rx, xr ∈ I.