ONLY FOR PROOF COPY IJMS, Vol. 11, No. 3-4, (July-December 2012), pp. 237-243 ON INTUITIONISTIC ANTI-FUZZY IDEAL IN RINGS P. K. Sharma & Vandana Bansal Abstract: In this paper, we apply the Yuanli, Zhang, Quanma’s idea of intuitionistic anti- fuzzy subgroups to subring and ideal of a ring. We introduce the notion of intuitionistic anti-fuzzy subring and intuitionistic anti-fuzzy ideal of rings, and investigate some related properties. Mathematics Subject classification: 03F55 , 08A72. Keywords: Intuitionistic anti-fuzzy subring (ideal), Upper �-level cut and lower �-level cut of a set, (�, �)-cut of a set, Anti image. 1. INTRODUCTION In this paper, we introduce the notion of intuitionistic anti fuzzy subring and intuitionistic anti-fuzzy ideal of rings, and investigate some related properties. 2. PRELIMINARIES In this section, we list some basic concepts and well known results of intuitionistic fuzzy set theory. Throughout this paper, R will be a commutative ring with unity. Definition (2.1): An intuitionistic fuzzy set A of a nonempty set R is an object of the form A = {< x, � A (x), � A (x) > : x � R}, where � A : R � [0, 1] and � A : R � [0, 1] are membership and non-membership functions such that for each x � R, we have 0 � � A (x) + � A (x) � 1. Remark (2.2): (i) When � A (x) + � A (x) =1, i.e., � A (x) = 1 – � A (x), then A is called fuzzy set (ii) We denote the IFS A = {< x, � A (x), � A (x) > : x � R} by A = (� A , � A ). Definition (2.3): If A is intuitionistic fuzzy subset of R, then the sets {< x, � A (x) > : x � R}, and {< x, � A (x) > : x � R}, are called fuzzy subset and anti-fuzzy subset of R with respect to intuitionistic fuzzy set A. For �, � � [0, 1], we define the following sets U 1 (A, �) = {x � R : � A (x) � �} and L 1 (A, �) = {x � R : � A (x) � �} and U 2 (A, �) = {x � R : � A (x) � �} and L 2 (A, �) = {x � R : � A (x) � �} 1 st International Conference on Mathematics and Mathematical Sciences (ICMMS), 7-8 July 2012. © Serials Publications ISSN: 0972-754X