Fuzzy Small Submodule and Jacobson L -Radical

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Volume 2011, Article ID 980320,12pages doi:10.1155/2011/980320

Research Article

Fuzzy Small Submodule and Jacobson L -Radical

Saifur Rahman

¹

and Helen K. Saikia

²

1Department of Mathematics, Rajiv Gandhi University, Itanagar 791112, India

2Department of Mathematics, Gauhati University, Guwahati 781014, India

Correspondence should be addressed to Helen K. Saikia,[email protected] Received 22 December 2010; Accepted 2 March 2011

Academic Editor: Enrico Obrecht

Copyrightq2011 S. Rahman and H. K. Saikia. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Using the notion of fuzzy small submodules of a module, we introduce the concept of fuzzy coessential extension of a fuzzy submodule of a module. We attempt to investigate various properties of fuzzy small submodules of a module. A necessary and suﬃcient condition for fuzzy small submodules is established. We investigate the nature of fuzzy small submodules of a module under fuzzy direct sum. Fuzzy small submodules of a module are characterized in terms of fuzzy quotient modules. This characterization gives rise to some results on fuzzy coessential extensions.

Finally, a relation between smallL-submodules and JacobsonL-radical is established.

1. Introduction

After the introduction of fuzzy sets by Zadeh 1, a number of applications of this fundamental concept have come up. Rosenfeld2was the first one to define the concept of fuzzy subgroups of a group. Since then many generalizations of this fundamental concept have been done in the last three decades. Naegoita and Ralescu 3 applied this concept to modules and defined fuzzy submodules of a module. Consequently, fuzzy finitely generated submodules, fuzzy quotient modules 4, radical of fuzzy submodules, and primary fuzzy submodules5,6were investigated. Saikia and Kalita7defined fuzzy essential submodules and investigated various characteristics of such submodules. These modules play a prominent role in fuzzy Goldie dimension of modules.

In this paper we fuzzify various properties of smallor superfluoussubmodules of a module. We define fuzzy small ephimorphism and fuzzy coessential extension of a fuzzy submodule. We investigate various characteristics of fuzzy small submodules. Necessary and suﬃcient conditions for fuzzy small submodules are established. We also investigate the nature of fuzzy small submodule under fuzzy direct sum. A relation regarding fuzzy small

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submodule of a module and fuzzy quotient module is also established. We attempt to fuzzify the well-known relation between the Jacobson radical and the small submodules of a module.

In8Basnet et. al. have shown that the relation between the Jacobson radical and the small submodules of a module does not hold in fuzzy setting whereas we have tried to achieve the relation. It is established that the JacobsonL-radical is the sum of all the smallL-submodules of a module. In case of a finitely generated module, the JacobsonL-radical is also a small L-submodule of the module under the condition thatL-{1}possesses a maximal element.

2. Basic Definitions and Notations

By R we mean a commutative ring with unity 1 and M denotes a R-module. The zero elements ofRandMare 0 andθ, respectively. A complete Heyting algebraLis a complete lattice such that for allA⊆ Land for allb∈ L,∨{a∧b |a ∈A} ∨{a| a∈ A}∧band

∧{a∨b|a∈A} ∧{a|a∈A}∨b.

Definition 2.1. A submoduleSof a moduleMover a ringRis said to be a small submodule ofMif for every submoduleNofMwithN /MimpliesSN /M.

The notationSMindicates thatSis a small submodule ofM.

Fuzzy set on a nonempty set was introduced by Zadeh 1in 1965. It is defined as follows.

Definition 2.2. By a fuzzy set of a moduleM, we mean any mappingμfromMto0,1. By 0,1^Mwe will denote the set of all fuzzy subsets ofM. Ifμis a mapping fromMtoL, where Lis a complete Heyting algebra thenμis called anL-subset ofM. ByL^Mwe will denote the set of allL-subsets ofM.

For each fuzzy setμinMand anyα∈ 0,1, we define two setsUμ, α {x∈ M| μx≥α},Lμ, α {x∈M|μx≤α}, which are called an upper level cut and a lower level cut ofμ, respectively. The complement ofμ, denoted byμ^c, is the fuzzy set onMdefined by μ^cx 1−μx. The support of a fuzzy setμ, denoted byμ^∗, is a subset ofMdefined by μ^∗{x∈M|μx>0}. The subsetμ∗ofMis defined asμ∗{x∈M|μx μθ}.

Definition 2.3see9. IfN⊆Mandα∈0,1^Mthenα_Nis defined as,

αNx

⎧⎨

⎩

α ifx∈N,

0 otherwise. 2.1

IfN{x}thenα{x} is often called a fuzzy point and is denoted by xα. Whenα 1 then 1_Nis known as the characteristic function ofN. From now onwards, we will denote the characteristic function ofNasχN.

Ifμ, σ∈0,1^Mthen

1μ⊆σ, if and only ifμx≤σx;

2 μ∪σx max{μx, σx}μx∨σx;

3 μ∩σx min{μx, σx}μx∧σx.

For any family{μ_i|i∈I}of fuzzy subsets ofM, whereIis any nonempty index set,

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4

i∈Iμix sup_i∈Iμix

i∈Iμix;

5

i∈Iμix inf_i∈Iμix

i∈Iμixfor allx∈M;

6 μσx ∨{μy∧σz|y, z∈M, yzx}.

Definition 2.4see9. LetXandYbe any two nonempty sets, andf:X → Y be a mapping.

Letμ ∈ 0,1^X andσ ∈ 0,1^Y then the imagefμ ∈ 0,1^Y and the inverse imagef⁻¹σ ∈ 0,1^Xare defined as follows: for ally∈Y

f μ y

⎧⎨

⎩

∨

μx|x∈X, fx y

, if f⁻¹ y /φ,

0, otherwise. 2.2

andf⁻¹σx σfxfor allx∈X.

Definition 2.5see9. Letζ∈0,1^Randμ∈0,1^M. Thenζμis a fuzzy subset ofMand it is defined by

ζμ x ∨

_n

i1

ζri∧μxi

|ri∈R, xi∈M, 1≤i≤n, n∈N, ⁿ

i1

rixix

2.3

for allx∈M.

Definition 2.6see9. A fuzzy setμofRis called a fuzzy ideal, if it satisfies the following properties:

1μx−y≥μx∧μy,

2μxy≥μx∨μy, for allx, y∈R.

The following definition is given by Naegoita and Ralescu3.

Definition 2.7 see3. Let M be a module over a ring R and L be a Complete Heyting algebra. AnLsubsetμinMis called anL-submodule ofM, if for everyx, y∈Mandr ∈R the following conditions are satisfied:

1μθ 1,

2μx−y≥μx∧μy, 3μrx≥μx.

We denote the set of allL-submodules ofMbyLM. IfL 0,1, thenμis called a fuzzy submodule ofM. The set of all fuzzy submodules ofMare denoted byFM.

Definition 2.8see9. Letμ, ν∈FMbe such thatμ⊆ν. Then the quotient ofνwith respect toμ, is a fuzzy submodule ofM/μ^∗, denoted byν/μ, and is defined as follows:

ν μ

x ∨{νz|z∈x}, ∀x∈ν^∗, 2.4 wherexdenotes the cosetxμ^∗.

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Definition 2.9see9. Letμ∈0,1^M. Then∩{ν|μ⊆ν, ν∈FM}is a fuzzy submodule of M, and it is called the fuzzy submodule generated by the fuzzy subsetμ. We denote this by μ, that is,

μ ∩

ν|μ⊆ν, ν∈FM

. 2.5

Letξ∈FM. Ifξμfor someμ∈0,1^M, thenμis called a generating fuzzy subset ofξ.

Remark 2.10. a If A is a nonempty subset of M, then χ_A χ_A, where A is the submodule ofMgenerated byA.

bIfx∈M, thenχRχ{x}is a fuzzy submodule ofMgenerated byχ{x}, and in this case,

χ_Rχ_{x}

χ_{x}

χ_{x}χ_Rx. 2.6

3. Preliminaries

This section contains some preliminary results that are needed in the sequel.

Lemma 3.1see10. LetMbe a module and suppose thatK≤N≤MandH≤M. Then aHKMif and only ifHMandKM;

bifKN, thenKM;

cifNis a direct summand ofM, thenKMif and only ifKN;

difM M1⊕M2 and K_i ≤ M_i fori 1,2, thenK1⊕K2 M1⊕M2 if and only if K1M1andK1M1.

Lemma 3.2see9. Letμ, ν∈FM. Thenμν∈FM.

Lemma 3.3see9. Letμ_i ∈ FM, for eachi ∈ I, where|I| > 1. Then

i∈I μ_i ∈ FMand

i∈Iμi

i∈Iμi.

Lemma 3.4see5. Letμ∈0,1^M. Then the level subsetμt{x∈M:μx≥t},t∈Imμis a submodule ofMif and only ifμis a fuzzy submodule ofM.

Corollary 3.5. μ∗is a submodule ofMif and only ifμis a fuzzy submodule ofM.

In the next two sections we present our main results.

4. Fuzzy Small Submodule

Definition 4.1. LetMbe a module over a ringRand letμ∈LM. Thenμis said to be a Small L-Submodule ofM, if for anyν ∈LMsatisfyingν /χ_Mimpliesμν /χ_M. The notation μLMindicates thatμis a smallL-submodule ofM.

IfL 0,1, thenμis called a fuzzy small submodule ofMand it is indicated by the notationμ_fM. It is obvious thatχ_{θ}is always a fuzzy small submodule ofM.

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Definition 4.2. Letμ ∈ FM. IfμfM, then we sayχM orMis a fuzzy small cover of χ_M/μ^∗orM/μ.

Definition 4.3. LetMandLbe any two modules over a ringR. Then an ephimorphismf : M → Lis called a fuzzy small ephimorphism, iff⁻¹χ{θ}fM.

It is obvious thatMis a fuzzy small cover ofM/μif and only if the canonical projection M → M/μ∗is a fuzzy small ephimorphism.

Definition 4.4. A fuzzy idealμofRwithμ0 1 is called a fuzzy small ideal of Rif it is a fuzzy small submodule of_RR.

Letμandσbe any two fuzzy submodules ofMsuch thatμ⊆σ, thenμis called a fuzzy submodule ofσ. Andμis called a fuzzy small submodule inσ, denoted byμ_fσ, ifμ_fσ^∗ in the sense that for every submoduleγinMsatisfyingγ_|σ^∗/χ_σ^∗impliesμ_|σ^∗γ_|σ^∗/χ_σ^∗by μ|σ^∗, γ|σ^∗we mean the restriction mapping ofμ,γonσ^∗resp..

Definition 4.5. LetMbe a module over a ringRand suppose thatμ, ν ∈ FMwithμ ⊆ ν.

Then we sayνlies aboveμinMorμis a coessential extension ofνifM/μis a fuzzy small cover ofM/ν, that is,ν/μfM/μχM/μ^∗.

Example 4.6. ConsiderM Z8 {0,1,2,3,4,5,6,7}under addition modulo 8. ThenMis a module over the ringZ. LetS{0,2,4,6}. Defineμ∈0,1^Mas follows:

μx

⎧⎨

⎩

1 if x∈S,

α otherwise. 4.1

where 0≤α <1. Thenμis a fuzzy small submodule ofM.

Remark 4.7. LetK {0,4}. ClearlyS, Kare the only proper submodules ofMandSK {0,2,4,6}{0,4}/M. ThereforeSM. Alsoμ∗{0,2,4,6}S. Moreover, if we takeα0 thenμbecomes the characteristic function ofS.

The above remark inspires us to state the following two theorems.

Theorem 4.8. LetMbe a module andN≤M. ThenNMif and only ifχNfM.

Proof. LetN M. We assumeχ_Nis not a fuzzy small submodule ofM. Thus there exists, ν∈FM,ν /χMsuch thatχNνχM. Letx∈M. Then

1 χNν

x ∨ χN y

∧νz|y, z∈M, yzx

. 4.2

So, there exist y0, z0 ∈ Mwith y0 z0 x such that χNy0∧νz0 1. Thus we have χNy0 1 andνz0 1, and soy0∈N,z0∈ν∗. This implies thatxy0z0∈Nν∗. Since x∈Mis arbitrary, so this impliesMNν_∗. ButN M. So, we must haveM ν_∗and this impliesνχM, a contradiction. Therefore,χNfM.

Conversely, we assumeχNfM. If possible letN be not a small submodule ofM.

Thus there exists T ≤ M, T /M, but N T M. Thus χ_N, χ_T ∈ FM and χ_N/χ_M,

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χT/χM. Let x ∈ M. SinceNT M, so there exist, n ∈ N,l ∈ T such thatx nl.

Now,

χNχT

x ∨ χN y

∧χTz|y, z∈M, yzx

≥χNn∧χTl 1. 4.3

This impliesχ_Nχ_Tχ_Mand it contradicts the fact thatχ_N_fM. HenceNM.

Here we present an alternative proof of the following theorem.

Theorem 4.9see8. Letμ∈FM. ThenμfMif and only ifμ∗M.

Proof. Suppose,μ_fM. LetN≤MandN /M. We claimμ_∗N /M. Now,N /Mimplies χN/χM. SinceμfM, so we must haveμχN/χM.

⇒there existsx0∈Msuch thatμχNx0<1,

⇒ ∨{μy∧χ_Nz|y, z∈M, yzx0}<1,

⇒eitherμy<1 orz /∈N, for ally, z∈M,yzx0,

⇒eithery /∈μ_∗orz /∈N, for ally, z∈M,yzx0,

⇒x0yz /∈μ∗N,

⇒μ∗N /M,

⇒μ∗M.

Conversely, we assumeμ_∗M. Letν∈FMbe such thatν /χ_M. This implies thatν_∗/M.

Thusμ∗ν∗/Msinceμ∗M. This implies that there existsx0∈Msuch thatx /∈μ∗ν∗. Thus for everyy, z∈Mwithyzx0implies eithery /∈μ∗orz /∈ν∗otherwisex0 ∈μ∗ν∗. Therefore,

μy<1 orνz<1 for everyy, z∈Mandyzx0,

⇒μy∧νz<1, for everyy, z∈Mandyzx0,

⇒ ∨{μy∧νz|y, z∈M, yzx0}<1,

⇒μνx0<1,

⇒μν /χ_M,

⇒μfM.

Corollary 4.10. Letμ, σ∈FM. Thenμfσif and only ifμ∗σ^∗.

Proof. Letμfσ ⇒ μfσ^∗. So, by above theorem we getμ∗ σ^∗. Conversely ifμ∗ σ^∗. So, by above theoremμ_fσ^∗. Henceμ_fσ.

Theorem 4.11see8. Letμ, ν∈FM. ThenμfM,νfMif and only ifμνfM.

As a consequence, we obtain the following.

Theorem 4.12. Any finite sum of fuzzy small submodules ofMis also a fuzzy small submodule in M.

Theorem 4.13. LetN≤Mandμ∈FMbe such thatμ⊆χ_N. Ifμ_|_N_fN, thenμ_fM.

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Proof. Letν∈FMbe such thatμνχM. Letx∈N. Then μ|N ν|N∩χNx,

∨{μy∧ν_|_N∩χ_Nz|y, z∈N, yzx}, ∨{μy∧νz|y, z∈N, yzx},

μνx 1sinceμνχ_M.

Therefore,μ_|_N ν_|_N∩χ_N χ_N. Sinceμ_|_N_fN, soν_|_N∩χ_Nχ_N. This implies thatχ_N⊆ν andμ⊆χN⊆ν. Thusμν⊆ν⇒χM⊆ν⇒χMν. HenceμfM.

Corollary 4.14. Letμ, ν∈FMandμ⊆ν. Ifμ_fν, thenμ_fM.

Proof. By definition,μ_fνmeansμ_fν^∗. Therefore, from above theorem we getμ_fM.

Theorem 4.15. Letμ, ν∈FM. Thenμfνif and only ifμ∗ ν∗.

Proof. Suppose, μ_fν. Let N ≤ ν_∗ andN /ν_∗. This impliesχ_N/χ_ν_∗. Since μ_fν, soμ χN/χν∗. This ensures that there existsx0inν∗such thatx0 ∈/μ∗N. Thusμ∗N /ν∗and henceμ∗ν∗.

Conversely, we assume μ_∗ ν_∗. This implies μ_∗ ν_∗ ≤ ν^∗. Therefore,μ_∗ ν^∗ Lemma 3.1b. So, byCorollary 4.10we haveμ_fν.

Definition 4.16. A fuzzy submodule σ in M is called a fuzzy direct sum of two fuzzy submodulesμandνifσμνandμ∩νχ_θ.

Definition 4.17. Anyμ∈FMis called a fuzzy direct summand ofMif there existsν∈FM such thatχMis a fuzzy direct sum ofμ, ν.

Theorem 4.18. Letμ, νbe fuzzy submodules of M withμ⊆νandνbe a direct summand ofM. Then μ_fMif and only ifμ_fν.

Proof. Suppose,μfM. Sinceνis a direct summand ofM, so there existsγ ∈FMsuch that

χMνγ, ν∩γχθ. 4.4

First we proveMν^∗⊕γ^∗. Now,ν∩γ χ_θimpliesν^∗∩γ^∗ θ. Also, fromχ_M νγwe haveM νγ^∗. We claimνγ^∗ν^∗γ^∗. For this letx∈νγ^∗. Then

νγx ∨{νy∧γz|y, z∈M, yzx}>0,

⇒νy>0 andγz>0 for somey, z∈M,yzx,

⇒xyz, for somey∈ν^∗,z∈γ^∗,

⇒x∈ν^∗γ^∗,

⇒νγ^∗⊆ν^∗γ^∗,

On the other hand ifx ∈ν^∗γ^∗, thenx yzfor somey, z∈Mwithνy >0,γz> 0.

This implies 0<∨{νy∧γz|y, z∈M, yzx} νγx. Thusx∈νγ^∗and so, we haveνγ^∗ν^∗γ^∗and henceM νγ^∗ν^∗γ^∗. Therefore,ν^∗is a direct summand

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ofM. But, we knowμfMif and only ifμ∗M. Thusμ∗ν^∗andν^∗is a direct summand ofMand so, byLemma 3.1cwe getμ_∗ν^∗. Hence, byCorollary 4.10we haveμ_fν.

The proof of the converse part follows fromCorollary 4.14.

Theorem 4.19. Letσ1, σ2∈FMandχMσ1⊕σ2. Also, letμ1, μ2∈FMbe such thatμ1⊆σ1

andμ2 ⊆ σ2. Thenμ1_fσ1 and μ2_fσ2 if and only ifμ1⊕μ2_fσ1⊕σ2, that is, if and only if μ1⊕μ2fM.

Proof. Suppose,μ1fσ1 and μ2fσ2. Thenμ1∗fσ₁^∗ andμ2∗fσ₂^∗Corollary 4.10. There- fore, we getμ1∗⊕μ2∗σ₁^∗⊕σ₂^∗Lemma 3.1d. Now,χ_Mσ1⊕σ2impliesM σ1⊕σ2^∗ σ^∗₁ ⊕σ₂^∗ and sinceμ1∗⊕μ2∗ μ1⊕μ2_∗, therefore we get μ1 ⊕μ2_∗ σ1⊕σ2^∗. Hence μ1⊕μ2fσ1⊕σ2Corollary 4.10.

Conversely, we assumeμ1⊕μ2σ1⊕σ2if and only ifμ1⊕μ2_fM. Now,μ1 ≤μ1⊕μ2

andμ1⊕μ2fMimplyμ1fM. Again, sinceσ1is a fuzzy direct summand ofMandμ1⊆σ1, so byTheorem 4.18we getμ1fσ1. Similarly, it can be proved thatμ2fσ2.

Corollary 4.20. LetM1≤M,M2≤MandMM1⊕M2. Letμ1∈FM1,μ2∈FM2. Define

μix

⎧⎨

⎩

μix if x∈Mi,

0 otherwise. 4.5

for allx∈M,i1,2. Thenμ1⊕μ2fMif and only ifμ1fM1andμ2fM2.

Theorem 4.21see8. LetM,Mbe any two modules over the same ring R and letf :M → M be a module homomorphism. IfμfM, thenfμfM.

Theorem 4.22. Letμ, ν ∈ FM be such thatμ ⊆ ν. Then ν_fM if and only if μ_fMand ν/μ_fχ_M/μ, that is, if and only ifμ_fMandν/μ_fM/μ^∗.

Proof. It is obvious that, χM/μ χM/μ^∗. So, it is suﬃcient to show νfM if and only if μ_fMandν/μ_fM/μ^∗.

Suppose,νfM. Then sinceμ ⊆ ν, soμfM. Next, we will proveν/μfM/μ^∗. Consider, the natural homomorphism,f : M → M/μ^∗, defined byfx x, wherex denotes the coset xμ^∗. Since ν_fM, so by Theorem 4.21 we get, fν_fM/μ^∗. Now, fνx

∨{νy|y∈M, fy x}, ∨{νy|y∈M, y x}, ∨{νy|y∈M, y−x∈μ^∗},

∨{νy|y∈M, y−xm, m∈μ^∗}, ∨{νxm|m∈μ^∗},

∨{νu|uxm, m∈μ^∗}, ∨{νy|u∈x},

ν/μxfor allx∈ν^∗.

Thusν/μfνonν^∗andfν_fM/μ^∗. Therefore,ν/μ_fM/μ^∗.

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Conversely, we assume μfMand ν/μfM/μ^∗. To show νfM, letσ ∈ FM be such thatσ /χ_M. Since μ_fM and σ /χ_M, so we have μσ /χ_M. This implies μ σ/μ /χM/μχM/μ^∗. Again, sinceν/μfM/μ^∗andμσ/μ /χM/μ^∗, so we must have

ν/μ μσ/μ /χ_M/μ^∗,

⇒νμσ/μ /χM/μ^∗,

⇒μνσ/μ /χM/μ^∗,

⇒νσ/μ∩νσ/χ_M/μ^∗,see9, Theorem 4.2.5

⇒νσ/μ /χM/μ^∗,sinceμ⊆ν

⇒νσ /χM,sinceχM/μχM/μ^∗. Therefore,ν_fM.

Corollary 4.23. Letμ, ν∈FMbe such thatμ⊆ν. IfνfM, thenμis a coessential extension of νinM.

Theorem 4.24. Letμ, ν∈FMbe such thatμ⊆ ν. Thenμis a coessential extension ofνinMif and only ifμσχMholds for allσ∈FMwithνσχM.

Proof. Suppose,μis a coessential extension ofν, that is,ν/μfχM/μ. If for allσ ∈ FM withνσχ_M, then

χM

μ νσ

μ νσμ

μ ν

μ σμ

μ . 4.6

So,χ_M/μ σμ/μsinceν/μ_fχ_M/μ. Thusμσχ_M.

Conversely, we assumeμσ χMholds for allσ ∈ FMwithνσ χM. If there existsσ ∈ FMcontainingμsuch thatν/μσ/μ χM/μ, then χM νσ. This yields χ_M μσ σ,sinceμ ⊆ σand soν/μ_fχ_M/μ. Henceμis a coessential extension of ν.

As a consequence, we obtain the following.

Theorem 4.25. Letγ, μ, ν∈FMbe such thatγ⊆μ⊆ν. Thenγis a coessential extension ofνin Mif and only ifμis a coessential extension ofνinMandγis a coessential extension ofμinM.

5. Jacobson L-Radical

Definition 5.1. Letμ ∈LM. Then μis called a maximalL-submodule ofMif μ /χMi.e., μis properL-submodule ofMand ifσany other properL-submodules ofMcontainingμ, thenμσ. Equivalentlyμis a maximal element in the set of all nonconstantL-submodules ofMunder point wise partial ordering.

The intersection of all maximalL-submodules ofMis known as JacobsonL-radical of Mand is denoted by JLRM.

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Theorem 5.2. Letμ∈L^M. Thenμis a maximalL-submodule ofMif and only ifμcan be expressed asμχ_μ_∗∪α_M, whereμ_∗is a maximal submodule ofMandαis a maximal element ofL-{1}.

Proof. Proof is similar to the proof of Theorem 3.4.3 of9and so it is omitted.

Lemma 5.3. Let M be a module over R and letx∈M. ThenχRχ{x}LMif and only ifχ{x}is in the sum of all smallL-submodules of M.

Proof. Suppose,χRχ{x}LM. Then byTheorem 4.9we have,χRχ{x}_∗ M. But, from Remark 2.10bwe have

χRχ{x}

χ{x}

χ{x}χRx. 5.1

Therefore,χ{x} ⊆ χRχ{x} as χ{x} ⊆ χRx. This implies χ{x} is in the sum of all smallL- submodules ofM.

Conversely, we assumeχ_{x}is in the sum of all smallL-submodules ofM. Thenχ_{x}⊆ μ_ifinite, whereμ_i_LMand so,x∈

μ_i_∗finite, μ_i_∗ M⇒x

x_ifinite, where xi∈μi_∗. Now,

χRχ{xi}

∗ χRxi

∗Rxi≤ μi

∗ 5.2

for alli. Therefore,χ_Rχ_{x_i_}_∗ Msinceμ_i_∗M. Therefore, byTheorem 4.9we have χRχ{xi}LM. This implies

χRχ{xi}finiteLM. But,χ{x} ⊆

χRχ{xi}finite, and so, we must have,χRχ{x}⊆

χRχ{xi}finite. Since

χRχ{xi}finiteLM. Therefore, we haveχ_Rχ_{x}_LM.

Definition 5.4 see9. LetL be a complete Heyting algebra. Then a ∈ L-{1} is called a maximal element, if there does not existc∈L-{1}such thata < c <1.

Theorem 5.5. For any moduleM, JLRM(the JacobsonL-radical ofM), is the sum of all small L-submodules ofM.

Proof. In view ofLemma 5.3, it is suﬃcient to show thatχRχ{x}LMif and only ifχ{x} ⊆ JLRM; or equivalently:χ_{x}is not a subset of JLRMif and only ifχ_Rχ_{x}is not a small L-submodule ofM. We will proof the later one.

Suppose,χ{x} is not a subset of JLRM. Then there exists a maximalL-submodule ν ofM such thatχ_{x} is not a subset ofν. This implies χ_Rχ_{x} χ_{x} which is not a submodule ofν. Sinceνis maximal, soν /χM. Therefore,νis a maximalL-submodule ofM which is properly contained inχRχ{x}ν. This impliesχRχ{x}νχM. Thusν /χMand χ_Rχ_{x}νχ_M. So, by definition we have,χ_Rχ_{x}is not a smallL-submodule ofM.

Conversely, we assumeχRχ{x}is not a smallL-submodule ofM. Then there exists a ν∈LMwithν /χMsuch thatχRχ{x}νχM. LetSbe the collection of all suchν. Then S / Φbecause,ν∈S. Now, for eachσ∈S,σ /χ_Mandχ_{x}is not a subset ofσ. Moreover, any

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proper fuzzy submodule containingνis also inS. Also,S,⊆forms a poset and the union of members of a chain inSis again a member ofS. Therefore, by Zorn’s lemma,Shas a maximal element,sayμ. Sinceμ∈Sso,χ{x}is not a subset ofμ. Now, letγ∈LMbe such thatμ⊆γ.

Ifγ /χM, thenγ is also inS. So, by maximality ofμ, we haveμ γ. This shows thatμis a maximalL-submodule ofM. Sinceχ_{x} is not a subset ofμ, so we must have χ_{x} is not a subset of JLRM.

Corollary 5.6. If JLRMis a smallL-submodule ofM, then it is the largest smallL-submodule of M.

Corollary 5.7. LetMbe finitely generated module, then JLRMexists and is a smallL-submodule ofMprovidedL-{1}has a maximal element.

Remark 5.8. However, if we takeL 0,1, thenL-{1}does not possess any maximal element, and so byTheorem 5.2, maximalL-submodule ofMdoes not exist. Since the existence of the JLRMdepends on the existence of maximalL-submodule ofM, therefore, the assumption of the existence of a maximal element ofL-{1}in corollary 5.5 is necessary.

Example 5.9. Let L {0,0.25,0.5,0.75,1}. Then L is a Complete Heyting algebra together with the operations minimummeet, maximumjoinand≤partial ordering, then 0.75 is a maximal element ofL-{1}. ConsiderMZ8 {0,1,2,3,4,5,6,7}under addition modulo 8.

ThenMis a module over the ringZ. LetS{0,2,4,6}. Defineμ∈0,1^Mas follows:

μx

⎧⎨

⎩

1 ifx∈S,

0.75, otherwise. 5.3

Thenμ∗{0,2,4,6}S, which is a maximal submodule ofZ8. Also,μχμ∗∪0.75M, where 0.75 is a maximal element ofL-{1}. So, byTheorem 5.2we haveμas a maximalL-submodule ofZ8. In fact,μis the only maximalL-submodule ofZ8and so JLRM μ. Sinceμ∗ Z8

and hence by Theorem 4.9, we get μLZ8. Thus JLRMLZ8. However, if we consider L 0,1, then 0,1does not have a maximal element and so byTheorem 5.2there does not exist any maximalL-submodulemaximal fuzzy submodule. So, JLRMdoes not exist whenL 0,1.

6. Conclusion

In this paper some aspects and properties of fuzzy small submodules have been introduced which dualize the notion of fuzzy essential submodules. This concept has opened a new avenue toward the study of fuzzy Goldie dimension, for example using the notion of fuzzy small submodules one can define hollow submodules and discrete submodules. In our future study we may investigate various aspects ofispanning dimension of fuzzy submodules, ii corank of fuzzy submodules,iiifuzzy lifting modules with chain condition on fuzzy small submodules, andivNoetherian and Artinian conditions on fuzzy Jacobson radical of a module.

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