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}

_{and Helen K. Saikia}

**2**

*1 _{Department of Mathematics, Rajiv Gandhi University, Itanagar 791112, India}*

*2 _{Department of Mathematics, Gauhati University, Guwahati 781014, India}*

Correspondence should be addressed to Helen K. Saikia,hsaikia@yahoo.com

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 small*L*-submodules and Jacobson*L*-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.

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 Jacobson*L*-radical is the sum of all the small*L*-submodules
of a module. In case of a finitely generated module, the Jacobson*L*-radical is also a small

*L*-submodule of the module under the condition that*L*-{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 of*R*and*M*are 0 and*θ*, respectively. A complete Heyting algebra*L*is a complete
lattice such that for all*A*⊆ *L*and for all*b*∈ *L*,∨{a∧*b* |*a* ∈*A}* ∨{a| *a*∈ *A}*∧*b*and

∧{a∨*b*|*a*∈*A}* ∧{a|*a*∈*A}*∨*b*.

*Definition 2.1. A submoduleS*of a module*M*over a ring*R*is said to be a small submodule

of*M*if for every submodule*N*of*M*with*N /M*implies*SN /M*.
The notation*SM*indicates that*S*is a small submodule of*M*.

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 mapping*μ*from*M*to0*,*1. By

0*,*1*M*we will denote the set of all fuzzy subsets of*M*. If*μ*is a mapping from*M*to*L*, where

*L*is a complete Heyting algebra then*μ*is called an*L*-subset of*M*. By*LM*we will denote the
set of all*L*-subsets of*M*.

For each fuzzy set*μ*in*M*and any*α*∈ 0*,*1, we define two sets*Uμ, α* {x∈ *M*|

*μx*≥*α}*,*Lμ, α* {x∈*M*|*μx*≤*α}*, which are called an upper level cut and a lower level
cut of*μ*, respectively. The complement of*μ*, denoted by*μc*, is the fuzzy set on*M*defined by

*μc*_{}_{x}_{1}_{−}_{μ}_{}_{x}_{}_{. The support of a fuzzy set}_{μ}_{, denoted by}* _{μ}*∗

_{, is a subset of}

_{M}_{defined by}

*μ*∗

_{}

_{{x}

_{∈}

_{M}_{|}

_{μ}_{}

_{x}_{}

_{>}_{0}

_{}}

_{. The subset}

_{μ∗}_{of}

_{M}_{is defined as}

_{μ∗}_{}

_{{x}

_{∈}

_{M}_{|}

_{μ}_{}

_{x}

_{μ}_{}

_{θ}_{}

_{}}

_{.}

*Definition 2.3*see9. If*N*⊆*M*and*α*∈0*,*1*M*then*αN*is defined as,

*αNx*

⎧ ⎨ ⎩

*α* if*x*∈*N,*

0 otherwise*.*

2.1

If*N*{x}then*α{x}* is often called a fuzzy point and is denoted by *xα*. When*α* 1
then 1*N*is known as the characteristic function of*N*. From now onwards, we will denote the

characteristic function of*N*as*χN*.
If*μ, σ*∈0*,*1*M*then

1*μ*⊆*σ*, if and only if*μx*≤*σx*;

2 *μ*∪*σx* max{μ*x, σx*}*μx*∨*σx*;

3 *μ*∩*σx* min{μ*x, σx*}*μx*∧*σx*.

4 * _{i∈I}μix* sup

_{i∈I}μix*;*

_{i∈I}μix5 * _{i∈I}μix* inf

*i∈Iμix*

*i∈Iμix*for all

*x*∈

*M*;

6 *μσx* ∨{μ*y*∧*σz*|*y, z*∈*M, yzx}*.

*Definition 2.4*see9. Let*X*and*Y*be any two nonempty sets, and*f*:*X* → *Y* be a mapping.

Let*μ* ∈ 0*,*1*X* and*σ* ∈ 0*,*1*Y* *then the imagefμ* ∈ 0*,*1*Y* *and the inverse imagef*−1*σ* ∈

0*,*1*X*are defined as follows: for all*y*∈*Y*

*f* *μ* *y*

⎧ ⎨ ⎩

∨*μx*|*x*∈*X, fx* *y,* if *f*−1 *y _{/}φ,*

0*,* otherwise*.* 2.2

and*f*−1*σx* *σfx*for all*x*∈*X*.

*Definition 2.5*see9. Let*ζ*∈0*,*1*R*and*μ*∈0*,*1*M*. Then*ζμ*is a fuzzy subset of*M*and

it is defined by

*ζμx* ∨

_{n}

*i1*

*ζri*∧*μxi*|*ri*∈*R, xi*∈*M,* 1≤*i*≤*n, n*∈*N,*

*n*

*i1*

*rixix*

2.3

for all*x*∈*M*.

*Definition 2.6*see9. A fuzzy set*μ*of*R*is called a fuzzy ideal, if it satisfies the following

properties:

1*μx*−*y*≥*μx*∧*μy*,

2*μxy*≥*μx*∨*μy*, for all*x, 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. An*L*subset*μ*in*M*is called an*L*-submodule of*M*, if for every*x, y*∈*M*and*r* ∈*R*
the following conditions are satisfied:

1*μθ* 1,

2*μx*−*y*≥*μx*∧*μy*,

3*μrx*≥*μx*.

We denote the set of all*L*-submodules of*M*by*LM*. If*L* 0*,*1, then*μ*is called a fuzzy
submodule of*M*. The set of all fuzzy submodules of*M*are denoted by*FM*.

*Definition 2.8*see9. Let*μ, ν*∈*FM*be such that*μ*⊆*ν*. Then the quotient of*ν*with respect

to*μ*, is a fuzzy submodule of*M/μ*∗, denoted by*ν/μ*, and is defined as follows:

_{ν}

*μ*

*x* ∨{ν*z*|*z*∈*x*}, ∀x∈*ν*∗_{,}_{}_{2.4}_{}

*Definition 2.9*see9. Let*μ*∈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 subset*μ*. We denote this by

μ, that is,

*μ*∩*ν*|*μ*⊆*ν, ν*∈*FM.* 2.5

Let*ξ*∈*FM*. If*ξ*μfor some*μ*∈0*,*1*M*, then*μ*is called a generating fuzzy subset of*ξ*.

*Remark 2.10.* a If *A* is a nonempty subset of *M*, then χ*A* *χ*A, where A is the

submodule of*M*generated by*A*.

bIf*x*∈*M*, then*χRχ{x}*is a fuzzy submodule of*M*generated by*χ{x}*, and in this
case,

*χRχ*{x}*χ*{x}*χ*{x}*χRx.* 2.6

**3. Preliminaries**

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

**Lemma 3.1**see10*. LetMbe a module and suppose thatK*≤*N*≤*MandH*≤*M. Then*

a*HKMif and only ifHMandKM;*

b*ifKN, thenKM;*

c*ifNis a direct summand ofM, thenKMif and only ifKN;*

d*ifM* *M1*⊕*M2* *and* *Ki* ≤ *Mi* *fori* 1*,2, thenK1*⊕*K2* *M1*⊕*M2* *if and only if*

*K1M1andK1M1.*

**Lemma 3.2**see9*. Letμ, ν*∈*FM. Thenμν*∈*FM.*

**Lemma 3.3**see9*. Letμi* ∈ *FM, for eachi* ∈ *I, where*|I| *>* *1. Theni∈I* *μi* ∈ *FMand*

_{i∈I}μi_{i∈I}μi.

**Lemma 3.4**see5*. Letμ*∈0*,*1*M. Then the level subsetμt*{x∈*M*:*μx*≥*t},t*∈Im*μis*

*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. LetM*be a module over a ring*R*and let*μ*∈*LM*. Then*μ*is said to be a Small

*L*-Submodule of*M*, if for any*ν* ∈*LM*satisfying*ν /χM*implies*μν /χM*. The notation
*μ*LMindicates that*μ*is a small*L*-submodule of*M*.

*Definition 4.2. Letμ* ∈ *FM*. If*μ*f*M*, then we say*χM* or*M*is a fuzzy small cover of

*χM/μ*∗or*M/μ*.

*Definition 4.3. LetM*and*L*be any two modules over a ring*R*. Then an ephimorphism*f* :

*M* → *L*is called a fuzzy small ephimorphism, if*f*−1*χ{θ}*f*M*.

It is obvious that*M*is a fuzzy small cover of*M/μ*if and only if the canonical projection

*M* → *M/μ∗*is a fuzzy small ephimorphism.

*Definition 4.4. A fuzzy idealμ*of*R*with*μ*0 1 is called a fuzzy small ideal of *R*if it is a

fuzzy small submodule of*RR*.

Let*μ*and*σ*be any two fuzzy submodules of*M*such that*μ*⊆*σ*, then*μ*is called a fuzzy
submodule of*σ*. And*μ*is called a fuzzy small submodule in*σ*, denoted by*μfσ*, if*μfσ*∗

in the sense that for every submodule*γ*in*M*satisfying*γ*|σ∗*/χ _{σ}*∗implies

*μ*

_{|σ}∗

*γ*

_{|σ}∗

*/χ*∗by

_{σ}*μ|σ*∗*, γ|σ*∗we mean the restriction mapping of*μ*,*γ*on*σ*∗resp..

*Definition 4.5. LetM*be a module over a ring*R*and suppose that*μ, ν* ∈ *FM*with*μ* ⊆ *ν*.

Then we say*ν*lies above*μ*in*M*or*μ*is a coessential extension of*ν*if*M/μ*is a fuzzy small
cover of*M/ν*, that is,*ν/μ*f*M/μχM/μ*∗.

*Example 4.6. ConsiderM* *Z8* {0*,*1*,*2*,*3*,*4*,*5*,*6*,*7}under addition modulo 8. Then*M*is a

module over the ring*Z*. Let*S*{0*,*2*,*4*,*6}. Define*μ*∈0*,*1*M*as follows:

*μx*

⎧ ⎨ ⎩

1 if *x*∈*S,*

*α* otherwise*.*

4.1

where 0≤*α <*1. Then*μ*is a fuzzy small submodule of*M*.

*Remark 4.7. LetK* {0*,*4}. Clearly*S, K*are the only proper submodules of*M*and*SK*

{0*,*2*,*4*,*6}{0*,*4}*/M*. Therefore*SM*. Also*μ∗*{0*,*2*,*4*,*6}*S*. Moreover, if we take*α*0
then*μ*becomes the characteristic function of*S*.

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

* Theorem 4.8. LetMbe a module andN*≤

*M. ThenNMif and only ifχN*f

*M.*

*Proof. LetN* *M*. We assume*χN*is not a fuzzy small submodule of*M*. Thus there exists,

*ν*∈*FM*,*ν /χM*such that*χNνχM*. Let*x*∈*M*. Then

1 *χNνx* ∨*χN* *y*∧*νz*|*y, z*∈*M, yzx.* 4.2

So, there exist *y0, z0* ∈ *M*with *y0* *z0* *x* such that *χNy0*∧*νz0* 1. Thus we have

*χNy0* 1 and*νz0* 1, and so*y0*∈*N*,*z0*∈*ν∗*. This implies that*xy0z0*∈*Nν∗*. Since

*x*∈*M*is arbitrary, so this implies*MNν*∗. But*N* *M*. So, we must have*M* *ν*∗and

this implies*νχM*, a contradiction. Therefore,*χN*f*M*.

*χT/χM*. Let *x* ∈ *M*. Since*NT* *M*, so there exist, *n* ∈ *N*,*l* ∈ *T* such that*x* *nl*.
Now,

*χNχTx* ∨*χN* *y*∧*χTz*|*y, z*∈*M, yzx*≥*χNn*∧*χTl* 1*.* 4.3

This implies*χNχTχM*and it contradicts the fact that*χNfM*. Hence*NM*.

Here we present an alternative proof of the following theorem.

**Theorem 4.9**see8*. Letμ*∈*FM. Thenμ*f*Mif and only ifμ∗M.*

*Proof. Suppose,μfM*. Let*N*≤*M*and*N /M*. We claim*μ*∗*N /M*. Now,*N /M*implies

*χN/χM*. Since*μ*f*M*, so we must have*μχN _{/}χM*.

⇒there exists*x0*∈*M*such that*μχNx0<*1,

⇒ ∨{μ*y*∧*χNz*|*y, z*∈*M, yzx0}<*1,

⇒either*μy<*1 or*z /*∈*N*, for all*y, z*∈*M*,*yzx0*,

⇒either*y /*∈*μ*_{∗}or*z /*∈*N*, for all*y, z*∈*M*,*yzx0*,

⇒*x0yz /*∈*μ∗N*,

⇒*μ∗N /M*,

⇒*μ∗M*.

Conversely, we assume*μ*∗*M*. Let*ν*∈*FM*be such that*ν /χM*. This implies that*ν*∗*/M*.

Thus*μ∗ν∗ _{/}M*since

*μ∗M*. This implies that there exists

*x0*∈

*M*such that

*x /*∈

*μ∗ν∗*. Thus for every

*y, z*∈

*M*with

*yzx0*implies either

*y /*∈

*μ∗*or

*z /*∈

*ν∗*otherwise

*x0*∈

*μ∗ν∗*. Therefore,

*μy<*1 or*νz<*1 for every*y, z*∈*M*and*yzx0*,

⇒*μy*∧*νz<*1, for every*y, z*∈*M*and*yzx0*,

⇒ ∨{μ*y*∧*νz*|*y, z*∈*M, yzx0}<*1,

⇒*μνx0<*1,

⇒*μν /χM*,

⇒*μ*f*M*.

* Corollary 4.10. Letμ, σ*∈

*FM. Thenμ*f

*σif and only ifμ∗σ*∗

*.*

*Proof. Letμ*f*σ* ⇒ *μ*f*σ*∗. So, by above theorem we get*μ∗* *σ*∗. Conversely if*μ∗* *σ*∗.

So, by above theorem*μfσ*∗. Hence*μfσ*.

**Theorem 4.11**see8*. Letμ, ν*∈*FM. Thenμ*f*M,ν*f*Mif and only ifμν*f*M.*

As a consequence, we obtain the following.

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

*Proof. Letν*∈*FM*be such that*μνχM*. Let*x*∈*N*. Then

*μ|N* *ν|N*∩*χNx*,

∨{μ*y*∧*ν*|N∩*χNz*|*y, z*∈*N, yzx}*,

∨{μ*y*∧*νz*|*y, z*∈*N, yzx}*,

*μνx* 1since*μνχM*.

Therefore,*μ*|N *ν*|N∩*χN* *χN*. Since*μ*|N*fN*, so*ν*|N∩*χNχN*. This implies that*χN*⊆*ν*

and*μ*⊆*χN*⊆*ν*. Thus*μν*⊆*ν*⇒*χM*⊆*ν*⇒*χMν*. Hence*μ*f*M*.

* Corollary 4.14. Letμ, ν*∈

*FMandμ*⊆

*ν. Ifμfν, thenμfM.*

*Proof. By definition,μ _{f}ν*means

*μfν*∗. Therefore, from above theorem we get

*μfM*.

* Theorem 4.15. Letμ, ν*∈

*FM. Thenμ*f

*νif and only ifμ∗*

*ν∗.*

*Proof. Suppose,* *μfν*. Let *N* ≤ *ν*∗ and*N /ν*∗. This implies*χN/χν∗*. Since *μfν*, so*μ*

*χN/χν∗*. This ensures that there exists*x0*in*ν∗*such that*x0* ∈*/μ∗N*. Thus*μ∗N /ν∗*and
hence*μ∗ν∗*.

Conversely, we assume *μ*∗ *ν*∗. This implies *μ*∗ *ν*∗ ≤ *ν*∗. Therefore,*μ*∗ *ν*∗

Lemma 3.1b. So, byCorollary 4.10we have*μfν*.

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

submodules*μ*and*ν*if*σμν*and*μ*∩*νχθ*.

*Definition 4.17. Anyμ*∈*FM*is called a fuzzy direct summand of*M*if there exists*ν*∈*FM*

such that*χM*is a fuzzy direct sum of*μ, ν*.

* Theorem 4.18. Letμ, νbe fuzzy submodules of M withμ*⊆

*νandνbe a direct summand ofM. Then*

*μfMif and only ifμfν.*

*Proof. Suppose,μ*f*M*. Since*ν*is a direct summand of*M*, so there exists*γ* ∈*FM*such that

*χMνγ,* *ν*∩*γχθ.* 4.4

First we prove*Mν*∗⊕*γ*∗. Now,*ν*∩*γ* *χθ*implies*ν*∗∩*γ*∗ *θ*. Also, from*χM* *νγ*we

have*M* *νγ*∗. We claim*νγ*∗*ν*∗*γ*∗. For this let*x*∈*νγ*∗. Then

*νγx* ∨{ν*y*∧*γz*|*y, z*∈*M, yzx}>*0,

⇒*νy>*0 and*γz>*0 for some*y, z*∈*M*,*yzx*,

⇒*xyz*, for some*y*∈*ν*∗,*z*∈*γ*∗,

⇒*x*∈*ν*∗_{}* _{γ}*∗

_{,}

⇒*νγ*∗⊆*ν*∗_{}* _{γ}*∗

_{,}

of*M*. But, we know*μ*f*M*if and only if*μ∗M*. Thus*μ∗ν*∗and*ν*∗is a direct summand
of*M*and so, byLemma 3.1cwe get*μ*∗*ν*∗. Hence, byCorollary 4.10we have*μfν*.

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 _{f}σ1*

*and*

*μ2*

_{f}σ2*if and only ifμ1*⊕

*μ2*⊕

_{f}σ1*σ2, that is, if and only if*

*μ1*⊕*μ2*f*M.*

*Proof. Suppose,μ1*f*σ1* and *μ2*f*σ2*. Then*μ1∗*f*σ*_{1}∗ and*μ2∗*f*σ*_{2}∗Corollary 4.10.

There-fore, we get*μ1∗*⊕*μ2∗σ*_{1}∗⊕*σ*_{2}∗Lemma 3.1d. Now,*χMσ1*⊕*σ2*implies*M* *σ1*⊕*σ2*∗

*σ*∗

1 ⊕*σ*2∗ and since*μ1∗*⊕*μ2∗* *μ1*⊕*μ2*∗, therefore we get *μ1* ⊕*μ2*∗ *σ1*⊕*σ2*

∗_{. Hence}
*μ1*⊕*μ2fσ1*⊕*σ2*Corollary 4.10.

Conversely, we assume*μ1*⊕*μ2σ1*⊕*σ2*if and only if*μ1*⊕*μ2fM*. Now,*μ1* ≤*μ1*⊕*μ2*

and*μ1*⊕μ2f*M*imply*μ1*f*M*. Again, since*σ1*is a fuzzy direct summand of*M*and*μ1*⊆*σ1*,
so byTheorem 4.18we get*μ1*f*σ1*. Similarly, it can be proved that*μ2*f*σ2*.

* 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,i*1*,2. Thenμ1*⊕*μ2*f*Mif and only ifμ1*f*M1andμ2*f*M2.*

**Theorem 4.21**see8*. LetM,Mbe any two modules over the same ring R and letf* :*M* → *M*

*be a module homomorphism. Ifμ*f*M, thenfμ*f*M.*

* Theorem 4.22. Letμ, ν* ∈

*FM*

*be such thatμ*⊆

*ν. Then*

*νfM*

*if and only if*

*μfMand*

*ν/μfχM/μ, that is, if and only ifμfMandν/μfM/μ*∗*.*

*Proof. It is obvious that,* *χM/μ* *χM/μ*∗. So, it is suﬃcient to show *ν*f*M* if and only if

*μfM*and*ν/μfM/μ*∗.

Suppose,*ν*f*M*. Then since*μ* ⊆ *ν*, so*μ*f*M*. Next, we will prove*ν/μ*f*M/μ*∗.
Consider, the natural homomorphism,*f* : *M* → *M/μ*∗, defined by*fx* *x*, where*x*
denotes the coset *xμ*∗. Since *νfM*, so by Theorem 4.21 we get, *fνfM/μ*∗. Now,

*fνx*

∨{ν*y*|*y*∈*M, fy* *x*},

∨{ν*y*|*y*∈*M,* *y* *x*},

∨{ν*y*|*y*∈*M, y*−*x*∈*μ*∗_{}}_{,}

∨{ν*y*|*y*∈*M, y*−*xm, m*∈*μ*∗_{}}_{,}

∨{ν*xm*|*m*∈*μ*∗_{}}_{,}

∨{ν*u*|*uxm, m*∈*μ*∗_{}}_{,}

∨{ν*y*|*u*∈*x*},

*ν/μx*for all*x*∈*ν*∗.

Conversely, we assume *μ*f*M*and *ν/μ*f*M/μ*∗. To show *ν*f*M*, let*σ* ∈ *FM*
be such that*σ /χM*. Since *μfM* and *σ /χM*, so we have *μσ /χM*. This implies *μ*
*σ/μ /χM/μχM/μ*∗. Again, since*ν/μ*f*M/μ*∗and*μσ/μ /χM/μ*∗, so we must have

*ν/μ* *μσ/μ /χM/μ*∗,

⇒*νμσ/μ /χM/μ*∗,

⇒*μνσ/μ /χM/μ*∗,

⇒*νσ/μ*∩*νσ/χM/μ*∗,see9, Theorem 4*.*2*.*5
⇒*νσ/μ /χM/μ*∗,since*μ*⊆*ν*

⇒*νσ /χM*,since*χM/μχM/μ*∗.

Therefore,*νfM*.

* Corollary 4.23. Letμ, ν*∈

*FMbe such thatμ*⊆

*ν. Ifν*f

*M, 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,μ*is a coessential extension of*ν*, that is,*ν/μ*f*χM/μ*. If for all*σ* ∈ *FM*

with*νσχM*, then

*χM*

*μ*

*νσ*

*μ*

*νσμ*

*μ*

*ν*

*μ*

*σμ*

*μ* *.* 4.6

So,*χM/μ* *σμ/μ*since*ν/μfχM/μ*. Thus*μσχM*.

Conversely, we assume*μσ* *χM*holds for all*σ* ∈ *FM*with*νσ* *χM*. If there
exists*σ* ∈ *FM*containing*μ*such that*ν/μσ/μ* *χM/μ*, then *χM* *νσ*. This yields

*χM* *μσ* *σ*,since*μ* ⊆ *σ*and so*ν/μfχM/μ*. Hence*μ*is 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 *μ*is called a maximal*L*-submodule of*M*if *μ /χM*i.e.,

*μ*is proper*L*-submodule of*M*and if*σ*any other proper*L*-submodules of*M*containing*μ*,
then*μσ*. Equivalently*μ*is a maximal element in the set of all nonconstant*L*-submodules
of*M*under point wise partial ordering.

The intersection of all maximal*L*-submodules of*M*is known as Jacobson*L*-radical of

* Theorem 5.2. Letμ*∈

*LM*

_{. Then}_{μ}_{is a maximal}_{L-submodule of}_{M}_{if and only if}_{μ}_{can be expressed}*asμχμ∗*∪*αM, 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}*LM

*if 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 small*L*
-submodules of*M*.

Conversely, we assume*χ*{x}is in the sum of all small*L*-submodules of*M*. Then*χ*{x}⊆

*μi*finite, where*μiLM*and so,*x*∈*μi*∗finite*,* *μi*∗ *M*⇒*xxi*finite, where
*xi*∈*μi*_{∗}. Now,

*χRχ{xi}*_{∗} *χRxi*_{∗}*Rxi*≤ *μi*_{∗} 5.2

for all*i*. Therefore,*χRχ*{xi}∗ *M*since*μi*∗*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 have*χRχ*{x}*LM*.

*Definition 5.4* see9. Let*L* be a complete Heyting algebra. Then *a* ∈ *L*-{1} is called a

maximal element, if there does not exist*c*∈*L*-{1}such that*a < c <*1.

**Theorem 5.5. For any module**M, JLRM(the JacobsonL-radical ofM), is the sum of all small

*L-submodules ofM.*

*Proof. In view of*Lemma 5.3, it is suﬃcient to show that*χRχ{x}*LMif and only if*χ{x}* ⊆

JLR*M*; or equivalently:*χ*{x}is not a subset of JLR*M*if and only if*χRχ*{x}is not a small
*L*-submodule of*M*. We will proof the later one.

Suppose,*χ{x}* is not a subset of JLR*M*. Then there exists a maximal*L*-submodule

*ν* of*M* such that*χ*{x} is not a subset of*ν*. This implies *χRχ*{x} χ{x} which is not a

submodule of*ν*. Since*ν*is maximal, so*ν /χM*. Therefore,*ν*is a maximal*L*-submodule of*M*
which is properly contained in*χRχ{x}ν*. This implies*χRχ{x}νχM*. Thus*ν /χM*and

*χRχ*{x}*νχM*. So, by definition we have,*χRχ*{x}is not a small*L*-submodule of*M*.

Conversely, we assume*χRχ{x}*is not a small*L*-submodule of*M*. Then there exists a

*ν*∈*LM*with*ν /χM*such that*χRχ{x}νχM*. Let*S*be the collection of all such*ν*. Then

proper fuzzy submodule containing*ν*is also in*S*. Also,*S,*⊆forms a poset and the union of
members of a chain in*S*is again a member of*S*. Therefore, by Zorn’s lemma,*S*has a maximal
element,say*μ*. Since*μ*∈*S*so,*χ{x}*is not a subset of*μ*. Now, let*γ*∈*LM*be such that*μ*⊆*γ*.
If*γ /χM*, then*γ* is also in*S*. So, by maximality of*μ*, we have*μ* *γ*. This shows that*μ*is a
maximal*L*-submodule of*M*. Since*χ*{x} is not a subset of*μ*, so we must have *χ*{x} is not a

subset of JLR*M*.

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

**Corollary 5.7. Let**Mbe 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, then*L*-{1}does not possess any maximal element,

and so byTheorem 5.2, maximal*L*-submodule of*M*does not exist. Since the existence of the
JLR*M*depends on the existence of maximal*L*-submodule of*M*, therefore, the assumption
of the existence of a maximal element of*L*-{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 of*L*-{1}. Consider*MZ8* {0*,*1*,*2*,*3*,*4*,*5*,*6*,*7}under addition modulo 8.
Then*M*is a module over the ring*Z*. Let*S*{0*,*2*,*4*,*6}. Define*μ*∈0*,*1*M*as follows:

*μx*

⎧ ⎨ ⎩

1 if*x*∈*S,*

0*.*75*,* otherwise*.*

5.3

Then*μ∗*{0*,*2*,*4*,*6}*S*, which is a maximal submodule of*Z8*. Also,*μχμ∗*∪0*.*75*M*, where

0.75 is a maximal element of*L*-{1}. So, byTheorem 5.2we have*μ*as a maximal*L*-submodule
of*Z8*. In fact,*μ*is the only maximal*L*-submodule of*Z8*and so JLR*M* *μ*. Since*μ∗* *Z8*
and hence by Theorem 4.9, we get *μ*LZ8. Thus JLR*M*LZ8. However, if we consider

*L* 0*,*1, then 0,1does not have a maximal element and so byTheorem 5.2there does
not exist any maximal*L*-submodulemaximal fuzzy submodule. So, JLR*M*does not exist
when*L* 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,

**References**

1 *L. A. Zadeh, “Fuzzy sets,” Information and Computation, vol. 8, pp. 338–353, 1965.*

2 *A. Rosenfeld, “Fuzzy groups,” Journal of Mathematical Analysis and Applications, vol. 35, pp. 512–517,*
1971.

3 *C. V. Naegoita and D. A. Ralescu, Application of Fuzzy Sets in System Analysis, Birkhauser, Basel,*
Switzerland, 1975.

4 *F. Z. Pan, “Fuzzy finitely generated modules,” Fuzzy Sets and Systems, vol. 21, no. 1, pp. 105–113, 1987.*

5 *F. I. Sidky, “On radicals of fuzzy submodules and primary fuzzy submodules,” Fuzzy Sets and Systems,*
vol. 119, no. 3, pp. 419–425, 2001.

6 *R. Kumar, S. K. Bhambri, and P. Kumar, “Fuzzy submodules: some analogues and deviations,” Fuzzy*

*Sets and Systems, vol. 70, no. 1, pp. 125–130, 1995.*

7 *H. K. Saikia and M. C. Kalita, “On fuzzy essential submodules,” Journal of Fuzzy Mathematics, vol. 17,*
no. 1, pp. 109–117, 2009.

8 *D. K. Basnet, N. K. Sarma, and L. B. Singh, “Fuzzy superfluous submodule,” in Proceedings of the 6th*

*IMT-GT Conference on Mathematics, Statistics and Its Applications, pp. 330–335, 2010.*

9 *J. N. Mordeson and D. S. Malik, Fuzzy Commutative Algebra, World Scientific, River Edge, NJ, USA,*
1998.

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