Int Jr. of Mathematics Sciences & Applications Vol.3, No.1, January-June 2013 Copyright  Mind Reader Publications ISSN No: 2230-9888 www.journalshub.com 11 ( ,) – Cut of intuitionistic fuzzy modules - II P.K. Sharma P.G. Department of Mathematics D.A.V. College, Jalandhar City , Punjab , India Abstract For any intuitionistic fuzzy set A = { < x , A(x) , A(x) > : x X} of a set X , the crisp subset C  ,  (A) = { x X : A(x)   , A(x)   } of X called the (, ) – cut of A is studied by the author in [6] and [7] when A is intuitionistic fuzzy subgroup of a group G and when A is intuitionistic fuzzy submodule of an R-module M respectively. Basnet in [4] studied the ( ,)- cut set when A is intuitionistic fuzzy ideal of a ring R. This paper is a continuation of author ’s earlier paper [7]. In this paper, some results concerning the sum and the product of two intuitionistic fuzzy submodules are obtained. A relationship between the ( ,)- cut set of sum and product of two intuitionistic fuzzy submodules with the ( ,)- cut sets of the two intuitionistic fuzzy submodules is established. Keywords: Intuitionistic fuzzy submodule (IFSM), (, ) – Cut set. Mathematics Subject Classification: 03F55 , 08A72 ,16D10 1. Introduction: The concept of intuitionistic fuzzy sets was introduced by Atanassov [1, 2] as a generalization of that of fuzzy sets and it is a very effective tool to study the case of vagueness. Further many researches applied this notion in various branches of mathematics especially in algebra and defined intuitionistic fuzzy subgroups, intuitionistic fuzzy subrings, and intuitionistic fuzzy sublattices, intuitionistic fuzzy submodules and so forth. 2. Preliminaries: In this section we recall some definitions and results which will be used later Definition (2.1)[1] Let X be a fixed non-empty set. An Intuitionistic fuzzy set (IFS) A of X is an object of the following form A = { < x , A(x) , A(x) > : x X}, where A : X  [0, 1] and A : X  [0, 1] define the degree of membership and degree of non-membership of the element x X respectively and for any x X , we have 0  A(x) + A(x)  1 . Remark (2.2)(i): When A(x) + A(x) = 1 , i.e. when A(x) = 1 - A(x) =  c A(x) . Then A is called fuzzy set.