Intuitionistic Fuzzy Small Submodules and Their Properties

The concept of an intuitionistic fuzzy set, which is a generalisation of fuzzy set, was introduced by K. T. Atanassov in 1986. In this paper using of intuitionistic fuzzy small submodules we get some results about this kind of fuzzy submodules. First we give some preliminary properties of intuitionistic fuzzy submodules. Then we attempt to investigate various properties of intuitionistic fuzzy small submodules. A necessary and sufficient condition for intuitionistic fuzzy small submodules is established. We investigate the nature of intuitionistic fuzzy small submodules under direct sum. Also we study on relation between intuitionistic fuzzy small submodules and level subsets of them, and get some interesting results in this sense.


Introduction
Fuzzy sets as a method of representing uncertainty in real physical world, is introduced by L. A. Zadeh [1] at first. Since then many authors have been applied this concept to introduce some classes of fuzzy sets.
In 1971 A. Rosenfeld [2] applied the notion of fuzzy sets to algebra and introduced fuzzy subgroups of a group. C. V. Negoita and D. A. Ralescu [3] applied this concept to modules and defined fuzzy submodules of a module. Then some classes of fuzzy submodules have been proceeded by some authors, such as finitely generated fuzzy submodules [4], primary fuzzy submodules [5], fuzzy essential submodules [6] and fuzzy small submodules [7].
In this paper after some essential preliminaries of intuitionistic fuzzy sets and submodules, we discuss on the class of intuitionistic fuzzy small submodules. We investigate various characteristics of intuitionistic fuzzy small submodules. Necessary and sufficient conditions for intuitionistic fuzzy small submodules are established. We also attempt to investigate well-known properties on intuitionistic small submodules. The nature of intuitionistic fuzzy submodules under fuzzy direct sums is investigated in this paper.
Throughout article R means a commutative ring with unity and M denotes a unitary left R-module. ∨ and ∧ denote respectively the maximum and minimum in the unit interval [0, 1].
Let X be a set. A map μ : X −→ [0, 1] is called a fuzzy subset of X. The complement of μ, denoted by μ c , is a fuzzy subset of X defined by μ c (x) = 1 − μ(x) for every x ∈ X.
is an intuitionistic fuzzy submodule of an R-module M, we write A is an IFM of M and denote by A ≤ IF M. In this case we say A is an intuitionistic fuzzy module too.
If α = 1, then μ α N = χ N and ν α N = χ c N , where χ N denotes the characteristic function of N. In this case we write Similar to this case we get if y ∈ N.
Now let x ∈ M and r ∈ R. If x ∈ N, then rx ∈ N and so we have So χ N (0) = 1 and hence 0 ∈ N. Now let x, y ∈ N and r ∈ R, then That is N is a submodule of M.
In general for for every t ∈ M define level subsets It is easy to see that

Proposition 2.9: Let M be an R-module and A
Proof: 1 and 2 follow from definitions and 3 follows from 1 and 2.
Let A ⊆ B be two IFM's of the module M. Define the quotient intuitionistic fuzzy module

Definition 2.10: Let M, N be two R-modules and
and

Intuitionistic Fuzzy Small Submodules
S. Rahman and H. K. Saikia introduced and studied the class of fuzzy small submodules as a class of fuzzy submodules [7]. In this section we will introduce intuitionistic fuzzy small submodules. We investigate various properties of this class of intuitionistic fuzzy submodules.
First recall the small submodules and give some properties of these submodules.
Proof: See Proposition 5.17, Lemma 5.18 and Proposition 5.20 of [22]. In the other hand by (ii) Since 0 is a small submodule in any module M, so χ IF 0 IF M by above proposition.
clearly. Then there exist a ∈ A * and k ∈ K such that

Proposition 3.7: Let M be a module and A = (μ
For the converse assume )| y, z ∈ N ; and y + z = x} and also In the next lemma we prove the modularity law in the lattice of intuitionistic fuzzy submodules of a module. Next we use this lemma to get some results about the intuitionistic fuzzy small submodules.

Lemma 3.13: Let M be a module and A
Proof: The first statement is clear.
In the other hand let x ∈ A * + B * . Then there exist y 0 ∈ A * and z 0 ∈ B * such that x = y 0 + z 0 . Now Hence x ∈ μ * (A+B) = (A + B) * . Therefore A * + B * ⊆ (A + B) * . The proof of (2) is similar to the proof of (1).
The next corollary immediately follows from Lemma 3.16 and Lemma 3.19.

Conclusion
The study of properties of intuitionistic fuzzy submodules of a module is a meaningful research topic in the theory of intuitionistic fuzzy sets. In this paper we focus our research on a special property of this kind of submodules namely intuitionistic fuzzy small submodules and study on some classical properties of them in details.

Disclosure statement
No potential conflict of interest was reported by the author(s).