On decomposition of intuitionistic fuzzy prime submodules

This article is in continuation of the first author’s previous paper on intuitionistic fuzzy prime submodules, [13]. In this paper, we explore the decomposition of intuitionistic fuzzy submodule as the intersection of finite many intuitionistic fuzzy prime submodules. Many other forms of decomposition like irredundant and normal decomposition are also investigated.


Introduction
Prime ideals play a central role in commutative ring theory. One of the natural generalizations of prime ideals which have attracted the interest of several authors is the notion of prime submodules (see for example [1,2,7,8] and [9]). These have led to more information on the structure of the R-module M . A proper submodule P of M is called prime if r ∈ R and x ∈ M , with rx ∈ P implies that r ∈ (P : M ) or x ∈ P , where (P : M ) = {r ∈ R : rM ⊆ N }, which is clearly an ideal of R. Also, if P is a prime submodule of M , then (P : M ) is a prime ideal of R.
The decomposition of an ideal (submodule) into prime ideal (prime submodule) is a traditional pillar of ideal (module) theory. It provides the algebraic foundation for decomposing an algebraic variety into its irreducible components. From another point of view, prime decomposition provides a generalization of the factorization of an integer as a product of prime-powers. In this paper, we study intuitionistic fuzzy prime decomposition, irredundant intuitionistic fuzzy prime decomposition and normal intuitionistic fuzzy prime decomposition.

Preliminaries
Throughout the paper, R is a commutative ring with unity 1, 1 = 0, M is a unitary R-module and θ is the zero element of M .
. Let X be a non-empty fixed set. An intuitionistic fuzzy set (IFS) A in X is an object having the form where the functions µ A : X → [0, 1] and ν A : X → [0, 1] denote the degree of membership (namely µ A (x)) and the degree of non-membership (namely ν A (x)) of each element x ∈ X to the set A, respectively, and Then A is called a fuzzy set.
(ii) The class of intuitionistic fuzzy subsets of X is denoted by IFS(X).
The following are two very basic definitions given in [5,6] and [11].
. Then A is called intuitionistic fuzzy ideal (IF I) of R if for all x, y ∈ R, the followings are satisfied   (ii) C · (A : C) ⊆ A; Theorem 2.7 ( [11]). For A i (i ∈ J), B ∈ IF S(M ) and C ∈ IF S(R). Then, we have:  Theorem 2.10 ( [11]). For A, B i ∈ IF S(M ) and C i ∈ IF S(R), (i ∈ J) . Then, we have: Definition 2.11 ( [4,12]). An intuitionistic fuzzy ideal P of a ring R, not necessarily nonconstant, is called intuitionistic fuzzy prime ideal, if for any intuitionistic fuzzy ideals A and B of R the condition AB ⊆ P implies that either A ⊆ P or B ⊆ P .  Theorem 2.15 ( [13]). (a) Let N be a prime submodule of M and α, β ∈ (0, 1) such that for all x ∈ M . Then A is an intuitionistic fuzzy prime submodule of M . (b) Conversely, any intuitionistic fuzzy prime submodule can be obtained as in (a).     An intuitionistic fuzzy prime decomposition A = ∩ n i=1 A i is called irredundant if no A i contains n j=1,j =i A j , and an irredundant intuitionistic fuzzy prime decomposition of A is called normal if the distinct A i have distinct associated intuitionistic fuzzy prime ideals. Next consider any one of the associated intuitionistic fuzzy ideal (A i : Then, we have: (A : B) = (( n k=1 A k ) : ( n j=1,j =i A j )) = n k=1 (A k : n j=1,j =i A j ), by Theorem 2.7. As n j=1,j =i A j ⊆ A j , ∀j, j = i implies (A j : n j=1,j =i A j ) = χ R , by Theorem 2.18. By the irredundancy of the set of A i , we have n j=1,j =i A j  Proof. We assume that A has an intuitionistic fuzzy prime decomposition ThenÁ i is an intuitionistic fuzzy prime submodule of M and (Á i : χ M ) = (A i : χ M ), by Corollary 2.17. Thus A =Á 1 ∩Á 2 ∩ · · · ∩Á m , where theÁ i have distinct associated intuitionistic fuzzy prime ideals. IfÁ i ⊇ m j=1,j =iÁ j , for some i, thenÁ i is deleted. Therefore A has a normal intuitionistic fuzzy prime decomposition.
Example 3.8. Let G be any finite abelian group of order n = p 1 p 2 . . . p k , where p i are distinct primes. Then by the structure theorem of finitely generated group we have . . , M k = x 1 , x 2 , . . . , x k = M be the chain of maximal submodules of M such that M 0 ⊂ M 1 ⊂ · · · ⊂ M k−1 ⊂ M k .