An Investigation of ℱ-Closure of Fuzzy Submodules of a Module

In this paper we introduce the notion of -closure of fuzzy submodules of a module M. Our attempt is to investigate various characteristics of such -closures. If is a non- empty set of fuzzy left ideals of R and μ is a fuzzy submodule of M then the -closure of μ is denoted by . If is weak closed under intersection then (1) -closure of μ exhibits the submodule character, (2) the intersection of – closures of two fuzzy submodules equals the -closure of intersection of the fuzzy submodules. If is weak closed under intersection then the submodule property of -closure implies that is left closed. Moreover, if is inductive then is a topological filter if and only if is a fuzzy submodule of M.


Introduction
Closure operators have played a vital role in a variety of areas of mathematics. In case of category of modules there are different closure operators like T-closed, T-honest, which have been studied by Fay and Joubert.These operators are useful in the study of various aspects of rings and modules. For example, Jara obtained the characterizations of rings of quotients using the honest operator. The theory of honest subgroups was developed by Abian and Rinehart in [1]. The concepts of isolated submodules, honest submodules are studied by Fay and Joebert, Jara in [2,3]. For a skew field, the notions of isolated submodules and honest submodules coincide. The honest submodules lead to a new characterization of ore domain. Moreover following the theory developed by Fay and Joebert, the in terms of In the category of groups isolated subgroups are useful in the study of torsion-free groups. The concept of super honest submodules was introduced by Joubert and Schoeman [4]. Super honest submodules of quasi injective modules are studied by Cheng [5]. In this paper our attempt is to extend to the notions of honest and superhonest submodules to fuzzy submodules. We define the concepts like honest fuzzy submodules, fuzzy closure, fuzzy torsion and superhonest fuzzy submodules. Various characteristics of honest and superhonest submodules are fuzzified in this paper.

Preliminaries
Throughout this paper R is a non commutative ring with unity and M is a left R-module. The zero elements of R and M are 0 and θ, respectively.

Definition 2.1:
A fuzzy subset μ of R is called a fuzzy left ideal if it satisfies the following properties [10]:

Definition 2.2:
A fuzzy subset μ of R is called a fuzzy ideal if it satisfies the following properties [10]:

Definition 2.3:
A fuzzy subset μ of M is called fuzzy submodules of M if the following conditions are satisfied: The set of all fuzzy submodules is denoted by F(M).

Definition 2.5:
Let μ be a fuzzy subset of a non-empty set X. Then a fuzzy point x t , x ∈ X, t ∈ (0, 1] is defined as the fuzzy subset x t of X such that x t (x) = t, and x t (y) = 0, for all y ∈ X − x. We use the notation x t ∈ μ if and only if x ∈ μ t .

Definition 2.6 ([7]):
Let Y be a subset of a non-empty set X and t ∈ (0, 1], then t Y is defined as When t = 1 then 1 Y , is known as the characteristic function of Y and it is denoted by χ Y .

Definition 2.7 ([8]):
A fuzzy left ideal μ of R is called a essential fuzzy left ideal of R, denoted by μ ⊆ e R, if for every nonzero fuzzy left ideal δ of R, there exist x( = 0) ∈ R such that x t ∈ μ and x t ∈ δ , for some t ∈ (0, 1].

Definition 2.8 ([8]):
Let μ and σ be two nonzero fuzzy left ideals of R such that μ ⊆ σ . Then μ is called a fuzzy essential left ideal in σ , denoted by μ ⊆ e σ if for every nonzero fuzzy left ideal δ of R satisfying δ ⊆ σ , there exist x( = 0) ∈ R such that x t ∈ μ and x t ∈ δ, for some t ∈ (0, 1].

Definition 2.13 ([9]):
Let μ be a fuzzy subset of an R-module M. Then the fuzzy subset ann(μ) of R is defined as follows:

Fuzzy Submodules
In this section let F be a non empty set of fuzzy left ideals of R. Definition 3.1: Let M be a left R-module and μ be a fuzzy submodule of M. Then we define fuzzy torsion of μ as follows: Also we define the F-torsion of μ as follows:

Definition 3.2:
Let M be a left R-module and μ be a fuzzy submodule of M. Then we define fuzzy closure of μ as follows: Also we define the F-closure of μ as follows: Hence the result follows.
The proofs of the following lemmas are similar.

weak closed under intersection, then F is left closed if and only if Cl M F (σ ) is a fuzzy submodule of M for any σ ∈ F(M).
. Hence the result follows.
(⇐). Let μ ∈ F and r t ∈ 1 R , then Cl R F (μ) = 1 R , hence r t ∈ Cl R F (μ) and therefore there is ν ∈ F such that νr t ⊆ μ. Theorem 3.14: Let F be an inductive set of fuzzy ideals, then the following statements are equivalent: (a) F is a topological filter.
is a fuzzy submodule for any σ ∈ F(M).
Proof: (a) ⇒ (b). F closed under intersection, so weak closed under intersection. It is given that F is left closed, hence by part (c) of the above theorem the result follows.
As the consequences of the Theorems 3.17, 3.18, 3.19 we obtain the following: boards for 25 mathematical journals. He has authored around 500 research papers, especially on algebra, algebraic hyperstructures and their applications and fuzzy logic. He has also published five books on algebra.