Extension of artin rees lemma for fuzzy module

In this article we introduce μ-filtered fuzzy module with a family of fuzzy submodules. It shows the relation between μ-filtered fuzzy modules and crisp filtered modules by level sets and some of it’s results. It is gained that the quotient on μ-filtered fuzzy module is too. After that we aim associated fuzzy graded module by μ-filtered fuzzy module Our goal is extension of important basic result about modules over a Noetherian ring known as Artin-Rees lemma to fuzzy term.


Introduction
Lotfi A. Zadeh [21] in 1965 introduced the notion of a fuzzy set.Rosenfield applied this concept in the group theory [18].In commutative algebra W. J. Liu [11] opened the way towards the development of fuzzy algebraic structures by introducing the notions of fuzzy normal subgroup, fuzzy subring and the product of fuzzy sets.Liu [20] introduced the notion of a fuzzy ideal of a ring.N. Kuroki [9] demonstrated the utility of the notion of the fuzzy set in the more general setting of semigroups.The concepts of fuzzy fields and fuzzy linear spaces were introduced by S. Nanda [16].Malik and Mordeson presented direct sum of fuzzy rings and fuzzy ideals [14].In commutative algebra the concept of fuzzy modules and L-modules were introduced by Negoita and Ralescu [17] and Mashinchi and Zahedi [15] respectively.In the recent years, fuzzy algebra are extended by Acar [2], Inan and Ozturk [8], Shao and Liao [19], Chen [4] and Gunduz [7].A filtered algebra is a generalization of the notion of a graded algebra.Examples appear in many branches of mathematics, especially in homological algebra and representation theory [1].The Artin-Rees lemma is a basic result about modules over a Noetherian ring, along with results such as the Hilbert basis theorem.It was proved in the 1950s in independent works by the mathematicians Emil Artin and David Rees.A special case was known to Oscar Zariski prior to their work.One consequence of the lemma is the Krull intersection theorem.The result is also used to prove the exactness property of completion [3].In this article we effort to introduce filtered fuzzy ring and µ-filtered fuzzy module in order to extend the subjects of commutative algebra.In the other part of the paper, the relationship between µ-filtered fuzzy module and its crisp form are investigated.We prove a fuzzy module is µ-filtered if and only if all its level sets be filtered module.In following it is gained that the quotient on µ-filtered fuzzy module is too.After that we aim associated fuzzy graded module by µ-filtered fuzzy module.In the following with the definition of µ-stable for fuzzy ideal µ, we express and prove the fuzzy version of Artin-Rees lemma.
Let R be an ordinary ring and M be a R-module.we adopt the concept of fuzzy modules, which was introduce by Negoita and Ralescu [17], as follows.(M, ν) is called a fuzzy R-module if the is a map satisfying the following conditions: Definition 2.1.[14] Let µ be a fuzzy subset of R. We define, Proposition 2.1.[14] Let µ be a fuzzy ring (fuzzy ideal), then µ * are subrings (ideals) of R.
Then ν/ζ is fuzzy module on ν * /ζ * and called the quotient of ν with respect to ζ .Definition 2.3.[13] Let f be a mapping from X into Y , and µ, ν fuzzy subsets on X,Y , respectively.The fuzzy subset f (µ) on y, defined by ∀y ∈ Y , Proposition 2.3.If µ be a fuzzy ring on R and ν is fuzzy module on M then µν is a fuzzy module.
Proof.It is clear that µν is fuzzy group by proof of proposition above.Now, for every x ∈ M and r ∈ R we have, This show that µν is a fuzzy module.
International Scientific Publications and Consulting Services Definition 2.6.[10] Let µ be a fuzzy ideal of R and let x ∈ R. Then the fuzzy subset x + µ of R defined by is termed as the fuzzy coset determined by x and µ.The set of all fuzzy cosets of µ in R is a ring under the binary operations and it is denoted by R/µ.We call it the fuzzy quotient ring of R induced by the fuzzy ideal µ.
Example 2.1.[6] Let I be an ideal in R and let R n = I n , n ≥ 0. Then {R n } is a filtration on R called I-adic filtration.
Definition 2.8.[6] Let R be a filtered ring.A filtered R-module M is an R-module M together with a family Example 2.2.[6] Let M be a filtered R-module and N an R-submodule of M. The filtration Definition 2.9.[5] Fuzzy ring µ on R is graded by natural number when fuzzy groups As a result of µ 0 µ 0 ⊆ µ 0 , µ 0 is a fuzzy ring of R.
iii. for all n, m ≥ 0,  This complete the proof.
We call the above fuzzy graded ring, associeted fuzzy graded ring by fuzzy ring.
Proof.The following statements show f gr(µ) is a fuzzy ring.First, by calculation gr(µ * ) = ⊕ n≥0 gr n (µ * ) is associeted grade ring by filtered ring µ * and we have In other hand This shows that since for every n ≥ 0, µ n µ n+1 is fuzzy group, it is clear that f gr(µ) is fuzzy group.The definition from f gr(µ) implies that be fuzzy graded ring.
Then f gr(ν) has a natural gr(µ * )-fuzzy graded module structure.This module is called the associated fuzzy graded module of ν.
Proof.By definition of f gr(ν), for every r It is obvious that f gr(ν) is fuzzy group and fuzzy graded.
Definition 3.3.Let ν be a µ-filtered fuzzy module on M with a filtration {ν n } and η an fuzzy ideal in R. The filtration is called η-stable if there exists some m such that for all n ≥ m, ην n = ν n+1 and ην n ⊆ ν n+1 for all n ≥ 0.
This complete the proof.Corollary 3.3.Let R be a Noetherian ring, µ an fuzzy ideal on R, M a finitely generated R-module and ζ ⊆ ν fuzzy modules on M. Then there exist some m such that µ m+k ν ∩ ζ = µ k (µ m ν ∩ ζ ) for all k ≥ 0.
Proof.Apply the fuzzy version of Artin-Rees lemma for the µ-adic filtration on ν.
Example 3.8.Let R be a Noetherian ring and µ an fuzzy ideal on R. Then we define filtration {ν n } n with ν n = µ n χ R for characteristic function χ R .Then the filtration is µ-stable and by 3.2 for every ideal I of R the fuzzy filtration induced by χ R on fuzzy ideal χ I is µ-stable and we have µ m+k χ R ∩ χ I = µ k (µ m χ R ∩ χ I ) for some m and for all k ≥ 0.

Conclusions
In [12] Murali and Makamba study the concepts of primary decompositions of fuzzy ideals and the radicals of such ideals over a commutative ring.Using such decompositions and a form of Nakayamas lemma, they prove Krulls intersection theorem on fuzzy ideals.In fact they show that if µ be a finitely generated fuzzy ideal of R such that µ is contained in the fuzzy Jacobson radical of R, Then ∧ ∞ n=0 µ n = 0.As mentioned before one consequence of the Artin-Rees lemma is the Krull intersection theorem and to prove the exactness property of completion.So with proving of fuzzy version of Artin-Ress lemma we can find a way for to extend Krull intersection theorem to fuzzy modules and apply that for processing of properties of fuzzy completion.

otherwise Definition 2.4. [14] Let µ and ν be two fuzzy subsets on R. Fuzzy subset µν on R is defined as following,
and ν be fuzzy rings (fuzzy ideal) on R. Then µν is fuzzy ring (fuzzy ideal) on R. Definition 2.5.Let µ be a fuzzy ring on R and ν is fuzzy module on M then µν is defined as Example 3.4.Let µ be a filtered fuzzy ideal by µ-adic filtration.Fuzzy module ν of M is µ-filtered with {ν n = µ n ν} that is called µ-adic filtration.Example 3.5.Let ν be a µ-filtered fuzzy module by {ν n } n≥0 in M and µ is fuzzy ring filtered by {µ n } n≥0 and ζ is a fuzzy submodule of ν.Then fuzzy module Corollary 3.1.Let ν be a µ-filtered fuzzy module on M. Then fuzzy submodule ζ is µ-filtered by induced filtration if and only if ζ x>α is a filtered module by induced filtration of ν x>α for every α ∈ [0, 1].Proof.It is enough we show ζ n = ζ ∩ ν n if and only if ζ x>α n = ζ x>α ∩ ν x>α n for all α ∈ [0, 1] and n ≥ 0. In fact, by this work we prove level sets of elements of induced filtration {ζ n } n≥0 are induced filtration of level sets ν.The proof of be filtered, gained by last proposition.If