84 

 This work is licensed under a Creative Commons Attribution 4.0 International License. 

 

  

 

 

Loc-hollow Fuzzy Modules with Related Modules 

 

                                                                                                           

 

 

Abstract 

     The concept of a small f- subm was presented in a previous study. This work introduced a concept 

of a hollow f- module, where a module is said to be hollow fuzzy when every subm of it is a small f- 

subm. Some new types of hollow modules are provided namely, Loc- hollow f- modules as a strength 

of the hollow module, where every Loc- hollow f- module is a hollow module, but the converse is not 

true. Many properties and characterizations of these concepts are proved, also the relationship 

between all these types is researched. Many important results that explain this relationship are 

demonstrated also several characterizations and properties related to these concepts are given.  

   Keywords:f- Modules, small f- subm, maximal f-subm, hollow f- modules, Loc- hollow f- modules. 

 

  𝟏.  𝐈ntroduction 

     Call M is L-hollow module where a module has a unique maximal submodule that contains all 

small submodules of M [10]. In this paper, we fuzzify these concepts from L-hollow module to Loc- 

hollow f- modules. Moreover, we generalize numerous properties of Loc- hollow f- modules.  This 

work contains four Parts. 

𝜤n part one, we recollect several 𝑑efinitions and 𝜌roperties that are useful are needed later. 

𝜤n part two, several fundamental ρroperties of L-hollow f- modules are argued.  

part three includes the Relation between hollow f- modules, and Loc- hollow f- modules. 

 Finally part four, we shall give the relation between Loc-hollow f- modules and different modules 

like as amply supplemented f- modules, indecomposable f- modules, and lifting f- modules 

1. Preliminaries: 

1.1 Definition [1] Let M ≠ ∅ , let Ɨ be the closed interval [0, 1] on the real line (real number), f- 

set A in M (a f- subset A of M) is fun from X to Ɨ.'' 

The following example describes the above definition: 

 

Ibn Al Haitham Journal for Pure and Applied Sciences 

Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index 

 

Doi: 10.30526/35.2.2731 

Article history: Received 21, November, 2021, Accepted,25 ,January, 2022, Published in April 2022. 

 

Shaheed Jameel Kharbeet 

Depatment of Mthmatics, College of Education of Pure 

Science, Ibn Al- Haitham, University of Baghdad, 

Baghdad – Iraq. 

Al- Maarif University College, Ramadi – Iraq. 

jkharbeet@uoa.edu.iq 

 

 

 

Hatam Yahya Khalf 

Depatment of Mthmatics, College of Education of Pure 

Science, Ibn Al- Haitham, University of Baghdad, 

Baghdad – Iraq. 

dr.hatamyahya@yahoo.com 

 

 

https://creativecommons.org/licenses/by/4.0/
mailto:kharbeet@uoa.edu.iq
mailto:dr.hatamyahya@yahoo.com


Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 
 

85 

1.2   Example [1] 

        Let M be the real line R and A be a fuzzy set of numbers that is much greater than I. Then one 

can accord an accurate characterization of A   specifying: 

A(x)   = {
1 −  

1

𝑥2
       𝑖𝑓    𝑥 > 1

0             𝑖𝑓   𝑥  ≤ 1
 ''      

1.3''Definition [2] 

  Let Xt: M → [0, 1] be a fuzzy set in M, where x ∈M, t ∈ (0,1] defined by  

Xt (y) = {
1    𝑖𝑓    𝑡 = 𝑦       

0    𝑖𝑓   𝑡 ≠ 𝑦  
∀y∈ M.  Xt is said a fuzzy singleton where x= 0, t = 1, therefor  

01 (y) =  {
 1    𝑖𝑓    𝑦 = 0       
0        𝑖𝑓   𝑦 ≠ 0 

,      is fuzzy singleton fuzzy zero singleton ''   

 

 

1.4  Definition [3]  

             Let u and p be two f- sets inΜ, then: 

1- (  u  ⋃ p  )(x)= max {u(x), p(x)}, ∀x ∈   M. 

2- (  u  ⋂ p )(x)= min {u(x), p(x)}, ∀ x ∈   M. (u ⋃ p ) a, (u  ⋂ p ) are fuzzy sets in      M  in 

general if {  ua , a ∈ A }, is u  family of fuzzy sets in M  , then  

((
⋂

a ∈ A
)𝑣𝑎) (x) = inf {ua (x), a ∈ A}, ∀ x  ∈  S. 

((
⋃

a ∈ A
)𝑣𝑎) (x) =   sup {ua (x), a ∈  A}, ∀x  ∈  S.'' 

1.5 Definition [5] [4]  

          If U is f- set  in M  ,for all  t ∈ (0,1].the set ut  = {  x ∈   M., u(x)  ≥  t }, is named a level 

subset  of u. Note that, ut is a subset of M in the ordinary sense.''   

1.6 Remark [1] 

           Let v, p are two f- sets inΜ, then: 

1-(u    ⋃   p   ) t = ut    ⋃   pt   }, for any t ∈ (0, 1]. 

2-(u   ⋂  p   )t = ut   ⋂  pt   for any t ∈ (0, 1]. 

3- U    =  p   if and only if ut   =  pt   for any t ∈ (0, 1].'' 

1.7 Definition [1]  

      Let U is f- sets inΜ, u is named empty fuzzy set, denoted by 𝜃 if   ⟺ u(x) =0, ∀ x∈ 𝑀.''      

1.8 D𝒅efinition [2] [4]  

   Let M be an Ř- module,  a  f- sets X of  M is named is   f- module where:  

1- A (0) = 1. 

2-A(x – y)  ≥   min {A (x), A(y)}. 

3-A (rx)  ≥   A (x), ∀x ∈   M, 𝛾 ∈ Ř. '' 

1.9  Definition [5][4] 

          Let A, B be two f- modules of an R- module M. B is named f- module of A, if A  ⊆ B.'' 

1.10 Definition [2]  

   Let A, C are f- subm of an f- module X, then A+C is   f- module m.'' 

1.11 Definition [7]      

 Let X be f- module, X is named simple f- module if X has only one proper f- subm,  

       which is 01.'' 

 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 
 

86 

1.12 Definition [12] 

        let A and B be two f- modules of an R-module X1 and X2. Let  𝑓 : A→B be a fuzzy 

homomorphism. If 𝜐 and 𝜎 are two f- subms of A, B, therefore  

1-   𝑓(𝜐) is fuzzy submodules of B, whenever  𝑓  is an epimorphism. 

2-𝑓-1(𝜎) is fuzzy submodules of A'' 

1.13 Definition [11] 

        Let 𝐴  be a proper f- sub of M. then 𝐴 is named a maximal fuzzy  submodule  of M. whether to 

any other    proper fuzzy  submodule  𝛽 of  M containing 𝐴 then 𝐴 = 𝛽.'' 

1.14 Definition [8] 

A proper fuzzy subm 𝐴 of an R-module X, is named a small fuzzy if 𝐴 is a fuzzy subm of 𝜒 , then 𝐴 

is named a small fuzzy in 𝜒 if every fuzzy submodule  𝛽  of   𝜒 , s.t  𝐴  + 𝛽  = 𝜒. Implies  𝛽  = 𝜒  '' 

1.15  Definition [13] 

 Let A be f- module of an R- module M.  A is named  finitely generated f- M odule  if there exists xt1 , 

xt2, , , , , , xt n  ⊆   A such that  A= { a1 (xt)t1  + a2 (xt)t2  + . . . + an (xn)tn}, where ai  ∈R , a(x)t  = (ax)t   

,  ∀ t  ∈ (0,1] ,  

 (Ax) t (y) = {
1    𝑖𝑓    𝑦 = 𝑎𝑥       
0       𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒  

      '' 

1.16  Definition [5]  

Let µ, 𝛶be two f- modules,  𝑓: X1 → X2 is a homomorphism between X1 and X2 respectively.Then F-

kernel of  𝑓 , F-kernel 𝑓 

Is the fuzzy subset of M1, defined by? 

F-ker 𝑓(x) = {
µ(0)    𝑖𝑓   𝑥 ∈ 𝑘𝑒𝑟 𝑓      

0       𝑖𝑓   𝑥 ∉ 𝑘𝑒𝑟 𝑓 
    '' 

1. Characterization of Loc- hollow fuzzy Module 

   Throughout the part, we introduce the definition of Loc- hollow f- modules and study the basic 

properties of these kinds of modules. 

2.1 Definition 

      An R-module X is Loc- hollow fuzzy module if X has a unique maximal fuzzy submodule that 

contains each a small fuzzy submodule of X. 

2.2 Example 

 Let M= Z4, R = Z,   define     

  X: M →  [  0,1]  𝑎𝑠 𝑓𝑜𝑙𝑙𝑜𝑤  

X(x) = {
1    𝑖𝑓    𝑥 ∈ 𝑀
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 

   and       Define   

 A: M →    [  0,1]  𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡   

A(x) = {
𝑡    𝑖𝑓    𝑥 ∈ 𝑁       

0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 
       , t ∈ (0, 1]  ,where N= 2Z4 

 Clear that x is a fuzzy module and At =N is a submodule in Xt , since At is only the maximal 

submodule of Xt by [9 ], therefore A is only the maximal fuzzy submodule of X and it is a fuzzy small 

submodule in X by [8 ] , which contains all fuzzy small submodule. On the other side   

1-  Let M=𝑍6 as   Z-module, R=Z. 

Define X: M⟶ [0, 1] By Xx) = {
1    𝑖𝑓    𝑥 ∈ 𝑀       

0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 
      

A: M⟶ [0, 1] by A(x) = {
𝑡    𝑖𝑓    𝑥 ∈   N     
0.75    𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 

     ∀  t ∈ (0,1]  , N=2Z 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 
 

87 

 B: M⟶ [0, 1] by B (t) = {
𝑡    𝑖𝑓    𝑥 ∈ 𝐿       

0,25    𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 
 ∀  t  ∈ (0,1] , L=3Z   

 Clear that X is fuzzy module and Xt=M, At=⟨2̅⟩, Bt =⟨3̅⟩ Aare submodules in Xt.  

But   At, Bt are two maximal submodules in Xt by [9], therefore A, B are two fuzzy maximal 

submodules in X . Thus X =Z6 is not a Loc-hollow fuzzy module. 

2.3 𝝆roposition  

       Let X be a f- module. Then X is Loc-hollow f- modules ⇔Xt is Loc-hollow f- modules ∀  t ∈

(0,1]. 

Proof:      

   ⟹  Suppose that X is a Loc-hollow f- modules, then there exists   a unique maximal fuzzy 

submodule A which contains all a small f -subm   in X. suppose that 𝑨 is  a f -subm   in 𝜒, therefore   

A+B=X for some f -subm   B in A of X, and (A+B)t=Xt ∀  t ∈ (0,1]  By [ 6] . But At is a unique 

maximal f -subm in   Xt, by [10]. Therefore At is a unique maximal submodule in Xt. Thus Xt is a 

Loc-hollow module ∀  t ∈ (0,1].  

⟸  Let Xt   be a Loc-hollow module, suppose that Bt be a submodule of Xt and (A+B)t=Xt for some 

f -subm   Bt in At  of Xt ∀  t ∈ (0,1]   , hence A+B=X or some f -subm   B   in A  of X. and Bt is a  f -

subm   in Xt by [8] . But A is a unique maximal f- subm of X which contains all a small f-subm by 

our assumption. Therefore X is Loc- hollow f- modules. 

(2.4)  Remark and Example  

1- Every Loc- hollow f- module is hollow f- module. 

Proof  

     Suppose that X is a Loc-hollow f- module, then there exists a unique maximal f- subm containing 

all a small f-subm say B in X. And since B is f -subm of X. Therefore B contain in X. By definition 

of hollow f- module. This implies that X is a hollow f- module.   

2- The convers Remark (2.4) (1) is not true in general for instant  

let M=Z p∞  . R=Z   , Define   X: M →    [  0,1]  𝑎𝑠 𝑓𝑜𝑙𝑙𝑜𝑤  

X(x) = {
1    𝑖𝑓    𝑥 ∈ 𝑀
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 

          

Clear that 𝜒 is f- module and Xt = Z p∞  = M. But Z p∞   has a small submodule by [5], and it is a 

fuzzy small by [1].Then Xt =𝑍𝑝
∞ is a hollow module. But not Loc- hollow module. Then X is not Loc- 

hollow module by Proposition: (2.3)  

The 3-Every local fuzzy module is Loc- hollow f- module, while the converse is not true in general. 

Proof:  

   Let M= Z2 ⊕  Q, R = Z,   define   by   

 X: M →  [  0,1]  𝑎𝑠 𝑓𝑜𝑙𝑙𝑜𝑤  

X(x, y) = {
1    ∀(𝑥, 𝑦)    𝑥 ∈ 𝑀

0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 
     such that x  ∈  0̅, y ∈ Q     ∀ t   ∈ (0,1]. 

Clear that 𝜒 is f- module and M=Xt , Z2 ⊕ Q is Loc- hollow module. But not local module since Xt = 

{0̅1} ⊕Q ≅ Q is a unique maximal submodule of Z2 ⊕ Q and {01}  ⊕ {01} is a small submodule of Z2  

⊕ Q and contained in {01} ⊕  Q ≅ Q. But Z2  ⊕ {01} is a proper submodule of Z2 ⊕ Q, also Z2 ⊕ 

{01} is not contained in {01} ⊕  Q. Thus X is Loc- hollow fuzzy module but a local fuzzy module. 

4-Every simple fuzzy submodule is not Loc- hollow fuzzy module. 

 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 
 

88 

Proof: 

Let M=Z5, R=Z,      

 𝜒 : M →  [  0,1]  𝑎𝑠 𝑓𝑜𝑙𝑙𝑜𝑤  

𝜒 (x) = {
1    𝑖𝑓    𝑥 ∈ 𝑀
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 

    , t ∈ (0, 1],   A(x) = {
𝑡    𝑖𝑓    𝑥 ∈ 𝑁       

0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 
 t ∈ (0, 1],   N=0̅. 

Clear that X is a simple fuzzy module. But not local fuzzy module since M= {0̅1} ⊕Z5 =Z5 is a unique 

maximal fuzzy submodule of Z5 and {01} is only one proper fuzzy submodule, which is a small submodule 

of Z5   and contained in {01} ⊕ Z5 =Z5. Also Z5 ⊕ {01} is not contained in {01} ⊕  Z5.That is X is a 

simple fuzzy module and not Loc- hollow fuzzy module. 

5-Every Loc- hollow f- module is not a simple fuzzy submodule: 

Proof: 

Let M= Z8   , R = Z,   Define   X: M →    [  0,1]  𝑎𝑠 𝑓𝑜𝑙𝑙𝑜𝑤  

X(x) = {
1    𝑖𝑓    𝑥 ∈ 𝑀
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 

                     and       Define    A: M →    [  0,1]  𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡  

A(x) = {
𝑡    𝑖𝑓    𝑥 ∈ 𝑁       

0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 
       , t ∈ (0, 1].   N= 2Z8    

 B: M →    [  0,1]  𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 

B(x) = {
𝑡    𝑖𝑓    𝑥 ∈ 𝐿       

0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 
       , t ∈ (0, 1].          L= 4Z8 

Consequently, X fuzzy module. and At =N , Bt =L   are two submodules of Xt and At a small 

submodule of Xt by [5 ], then A is a  small submodule of  X by [ 9], Which is a maximal submodule 

of       𝜒    . But   (   ⊕ 𝑩)t  =Xt by [10] are two direct sum submodules of Xt. and A ⊕ B = 𝜒 .This 

implies that  X is not a simple f- module. 

 2.5 𝝆roposition:  

        Epimorphic image of Loc- hollow f- module is Loc- hollow f- module. 

𝝆roof: 

  Let X1 be Loc- hollow f- module and let f: X1 →  X2 be an epimorphism with X2 is a f- module [9]. 

Suppose that At be a unique maximal submodule of X2 with At + Ct =( X2)t where Ct proper submodule 

of (X2)t =Xt,  ∀ t ∈ (0, 1], but A is a   unique maximal submodule [10 ].Then A + C = X2. Hence At+ 

Ct =(X1)t .Now. f
-1 (A) is a  unique maximal fuzzy submodule of X1 since otherwise  f

-1 (A)= X1= Xt 

,  ∀ t ∈ (0, 1] and hence f (Ωf-1 (A)  )= f  (X1) = X2 implies that  A= X2 , which is a contradiction, with 

A is a unique maximal fuzzy submodule of X2, thus  f
-1 (A) is a unique maximal fuzzy submodule of 

X1 and since X1 is Loc- hollow fuzzy module, therefore f
-1 (A) contains all a fuzzy small submodule 

of X1 and hence f (f
-1 (A)) is a fuzzy small submodule of f  (X1) . This A is a fuzzy small submodule 

of X2, where At is an f small submodule of (X2 )t, ∀ t ∈ )0, 1]. Therefore X2 is Loc- hollow fuzzy 

module. 

The next proposition appears more particulars of Loc- hollow fuzzy modules. 

2.6 Proposition:  

       Let C be a small Ƒuzzy submodule of f- modules X. whether 𝜒/C is Loc-hollow f- modules then 

X is Loc- hollow f- modules. 

 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 
 

89 

Proof 

          Let X/C is Loc- hollow f- modules, where C is a small Ƒuzzy submodule of X. Then there exists 

a   unique maximal small Ƒuzzy submodule A/C in   𝜒 /C, with X=B+ D and Xt=Bt+ Dt,  ∀  t ∈ (0,1] 

by [6].  Where B is a small Ƒuzzy submodule of 𝜒  and D is a proper Ƒuzzy submodule of   𝜒X. then  

(D+𝑩)/C= 𝜒 /C implies that( (D+C/C)) + (B+C)/C= 𝜒 /C, since  (D+C/C) is a Ƒuzzy submodule of  

A/C ,  𝜒 /C is Loc-hollow f- modules ,then  (D+C/C) is a small fuzzy of  X/C. Thus   (B+C)/C =X/C, 

so B+C=X. Since Ct is a submodule of Xt, Bt =Xt by [9]. Therefore C is a f- subm of X, and B=X. 

Thus, X is Loc- hollow f- modules. 

2.7 Corollary:  

         Suppose that 𝜒 be f- modules. If X be a   Loc-hollow f- modules then   𝜒 /𝑨 is Loc- hollow f- 

modules for every proper f- subm 𝑨 of𝜒.  

 

Proof: 

       Suppose that 𝜒 is Loc-hollow f- Modules. Then there exists a   unique maximal f- subm A contains 

all a small f- subm. Let 𝑨 be a proper Ƒuzzy submodule of Loc-hollow f- modules 𝜒 and let   𝜋: 𝜒 →  

𝜒 /𝑨 be nature epimorphism then 𝜒 /𝑨 is Loc-hollow f- modules by proposition (2.5). 

2.8   𝝆roposition:  

      Let A be a proper f- subm of an R-module X. If X is Loc-hollow f- module and X/A is finitely 

generated then X is finitely generated f- module. 

Proof: 

            Let A be a proper fuzzy submodule of Loc-hollow fuzzy  module X with  X/A is finitely 

generated then X/A=a1( (xt)t1) +A  )  + a2( (xt)t2 +A) + . . . + an ((xn)tn +A)}, where ai  ∈R and a(x)t  = 

(ax)t   ,  ∀ t  ∈ (0,1] , xi ∈  X,  for all i= 1,2, , ,n  , where  

 (ax)t (y) = {
1    𝑖𝑓    𝑦 = 𝑎𝑥       
0       𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒  

              ∀ t∈ (0,1] .  

We claim that X =a1( (xt)t1)  + a2( (xt)t2 + . . . + an ((xn)tn  ∀ ai  ∈R. Let x ⊆X ,then x +A ∈ X/A , implies 

that  C+ A= a1(x1+A)+ a2(x2)+A +. . . + an(xn +A)= a1(x1)+ a2(x2) +. . . + an(xn )+A . Thus implies that 

c= a1(x1) + a2(x2) + . . . + an (xn) +u for some u ⊆  A. Thus X = a1(x1)+ a2(x2) +. . . + an(xn )+u , and 

since X is Loc-hollow f- modules., therefore X is hollow fuzzy module by remark( (2.1.4)(1) ). Then 

A is a small f- subm of X which implies that X= a1(x1) + a2(x2) + . . . + an (xn) where ai ∈R. Thus X 

is finitely generated f- modules. 

3. Relation between Hollow fuzzy module and Loc- Hollow fuzzy modules 

     We introduce the following definition let 𝜒 be an f- module of an R-Module X.  X is named Hollow 

f -  module If every proper f- submodule of X is a small f- module of X. Hence we can say every Loc- 

hollow fuzzy module is Hollow fuzzy module, and we introduced an examples display that the 

converse is not true. In this section, we discuss conditions under which Hollow fuzzy modules could 

be Loc- hollow fuzzy modules. 

 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 
 

90 

 3.1 Proposition:  

         Let X be a f- module of an R-module, X is a Loc –hollow f- module ⟺X is a Hollow and cyclic 

f- module.  

Proof; 

     ⟹   Suppose that X is Loc –hollow f- modules then   there exists a unique maximal fuzzy 

submodule A which contains all a small f- submodule of X.  This mean   there exists At is a small 

submodule of Xt,  ∀ t∈ (0,1]. Let xt ⊆X   with xt  ⊈ A, (xt) is a submodule of Xt. then Rx is a f- 

submodule of X. We claim that X = (xt) such that ys ⊆ X has written ys= xt 𝑎ℓ for some fuzzy singleton  

𝑎ℓ  of R where t, s   , ℓ  ∈   (0, 1] , Rx = X  , So Rxt = Xt ,  ∀ t∈ (0,1].If  Rx ≠X  then Rx is proper a 

small f- submodule of X and hence Rx is a small fuzzy submodule of A which implies that x ∈A , 

which contradiction. Thus Rx =X   then X is cyclic modules. Now since X Loc –hollow f- modules 

then X is Hollow f- modules by remark (2.4) (1).  

⟸  suppose that X is hollow cyclic f- modules and then it is finitely generated f- modules and hence 

X has maximal f- submodule suppose that X is hollow cyclic f- modules and then it is fuzzy finitely 

generated module and hence X has maximal f- submodule contained all proper a small f- submodule 

say A, Let B f- submodule of X. If B is not contained in A, then B+A =X, but X is a Loc –hollow f- 

modules. Thus A =X hence At =Xt where t ∈ (0, 1] by [8], which contradiction. This implies that 

every proper f- submodule of X is contained in A, which implies that X has a unique maximal f- 

submodule that contained all proper f- submodules of X. Therefore X is a local f- module.   

3.2 Corollary:  

         Let X be a f- Modules an Ř – module. X is a Loc- hollow f- module ⟺ X is a hollow and finitely 

generated f- module. 

Proof: 

⟹    Suppose that X is Loc- hollow f- modules, then X is hollow f- modules and cyclic by 

proposition (3.1), and since X is cyclic f- modules,. Thus X is finitely generated 

 ⟸  let  X finitely generated hollow module the  X = { Ra1 (xt)t1  + Ra2 (xt)t2  + . . . + Ran (xn)tn}, 

where ai  ∈R , a(x)t  = (ax)t   ,  ∀ t  ∈ (0,1] .If   X ≠   Ra1 (xt) t1 then   Rxt1   is proper fuzzy 

submodule of X which implies that R xt1 is fuzzy small submodule of X. hence X = {Ra1 (xt) t1 + 

Ra2 (xt) t2 + . . . + Ran (xn) tn}.So, we delete the summand one by until we have X= Ra1 (xt) ti    for 

some i .thus X is cyclic f- Modules and by   proposition (3.1). Therefore X is Loc- hollow f- 

modules. 

 

3.3 𝝆roposition:  

         Suppose that 𝜒 be f- modules of an 𝑅 – module   M, 𝜒 is a Loc- hollow f- module⟺ X is hollow 

and has a unique maximal fuzzy submodule. 

Proof 

      ⟹      Suppose that X is Loc- hollow fuzzy module then X is hollow a f- modules by remark (2.4) 

(1). And by definition (2.1), then X has a unique maximal fuzzy submodule fuzzy module.  

⟸   Let X be t hollow f- modules which have unique maximal fuzzy submodule fuzzy module, say A, 

we only have to show that X is a cyclic fuzzy module, let xt ⊆ X and xt ⊆ A clear that x ∈ Xt  and x 

 ∉ At, ∀  ∈ (0,1], then Rxt + At=Xt   و  ∀ t  ∈ (0,1].  But  𝜒 is a hollow f- module s then 𝑨 small fuzz 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 
 

91 

submodule of 𝜒 and hence X= Rx. Therefore X is cyclic fuzzy module, and by proposition (3.1). 

Therefore X is Loc- hollow f- modules. 

3. 4 𝝆roposition 

         Let X be f- module, X is a Loc- hollow f- module ⟺X is cyclic and X/A is indecomposable. 

Proof: 

          ⟹  Suppose that X is Loc- hollow f- module then X is hollow and cyclic f- module by proposition 

(2.1). Then X/A indecomposable. 

 ⟸   Let X be a cyclic f- module and every X/A is indecomposable, then by part one.  X is hollow f- 

module and proposition (2.1). Thus X Loc- hollow f- module. 

3.5 Proposition 

         Let X be a fuzzy module of an R – module M, X is a Loc- hollow f- module⟺ X is a hollow f- 

module, and F-Rad(X)≠ X. 

Proof: 

   ⟹  Suppose that X is Loc- hollow f- module, then X is X hollow and cyclic f- module by 

𝜌proposition (2.1). Xt is cyclic module and it’s finitely generated, ∀ t  ∈ (0,1] by [9]. Hence F-

Rad(X)≠ X. 

⟸  Let X hollow fuzzy module and F-Rad(X)≠ X is a small f- submodule of X. F-Rad(X) is unique 

maximal f- submodule of X and this X/ Rad(X) is a simple fuzzy module and hence cyclic. Implies 

that X/ Rad(X) = <x +Rad(x)> for some x ⊆ X.  

We clam that X= Rx. Let u ∈  X then u+ Rad(x) ∈ X/ Rad(X), therefore a ∈ R such that u+ Rad(x) = 

a (u+ Rad(x)) = au+ Rad(x), implies that u- ax ∈  Rad(x) which implies that u- ax = B for some B ∈  

Rad(x). Thus u= ax   + B ∈  Rx +Rad(x), hence X= Rx +Rad(x). But F-Rad(x) is fuzzy small 

submodule of X implies that X= Rx. Thus X is cyclic f- module by 𝜌roposition (2.1).we get X is a 

Loc- hollow f- module. 

4.  The Relationships between Loc- Hollow Fuzzy Module and Other Types of Modules 

          In this section, we shall give the relation between the Loc-hollow fuzzy module and different 

modules like as amply supplemented modules, indecomposable modules and lifting modules. 

We shall fuzzify the following definitions: 

    4.1 Definition: 

let   A, B are f- submodules of fuzzy module X  Then A is named a fuzzy supplement of B in X , if 

A is minimal with A + B = X. equivalently, A is named f- supplement of B  ⟺ 𝑨 + 𝑩= 𝜒 and 𝑨 ∩ 𝑩 

is a small f- subm of  𝑨.  

An f- subm A of X is named f- supplement, if there is f- subm B of X such that A is f-supplement of 

B. 

 

4.2 Example 

    Let M =Z4    , Define X: M → [0, 1]   and define: 

A:  M → [0, 1], define by   

A (t) = {
𝑡    𝑖𝑓    𝑥 ∈ 𝑁       

0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 
        ∀  t ∈ (0, 1]   , N =2Z    

And let   B:  M → [0, 1], define by   



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 
 

92 

B (t) = {
𝑡    𝑖𝑓    𝑥 ∈ 𝐿     

0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 
        ∀  t ∈ (0, 1], L=⟨0̅⟩      

  Clear that At = ⟨2̅⟩ , Bt = ⟨4̅⟩  are two submodules of 𝜒 t , 𝜒 t = M = Z4. Hence (A+B)t = (𝜒)t   of 𝜒 t ,  

∀  t ∈ (0, 1] by[3 ],therefor A+B = 𝜒  ∀ A,B  𝛼re a f- subm of  𝜒. But Bt is a supplement in At of 𝜒 t 

by [9], hence B= 01 of 𝜒 is f- supplement in A of𝜒. On the other side, A is not f- supplement because 

01 is a minimal in 𝜒 =Z4. 

  4.3 Definition:  

   Let A, B are fuzzy submodules of f- module X. Then X   is called amply f- supplemented with A + B 

= X if there is a supplement u of A such that u ⊆ B in X. 

  4.4 Example 

Let   M=Z12 as Z-module, define  

X: M  →   [  0,1]  𝑎𝑠 𝑓𝑜𝑙𝑙𝑜𝑤  ,   X(x) = {
1    𝑖𝑓    𝑥 ∈ 𝑀
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 

        Define   

  A: M →    [  0,1]  𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡  A(x) = {
𝑡    𝑖𝑓    𝑥 ∈ 𝑁       

0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 
, t ∈ (0, 1].  N= 3Z   Also   

B: M →    [  0,1]  𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡          B(x) = {
𝑡    𝑖𝑓    𝑥 ∈ 𝐿       

0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 
   , t ∈ (0, 1], L= 2Z    

C: M →    [  0,1]  𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡          B(x) = {
𝑡    𝑖𝑓    𝑥 ∈ 𝐾       

0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 
   , t ∈ (0, 1], K= 4Z    

Clear that X is fuzzy module  and Xt =M  ,At =3Z , Bt=2Z and Ct=4Z are  submodules in Xt  with  (A 

+ B)t=( 𝜒)t  , ∀  t ∈ (0, 1] ,therefore 𝑨 +  𝑩= 𝜒  where 𝑨 , 𝑩 are fuzzy submodules of X  . But 𝑨 is 

fuzzy supplement of B in X .Also   we have A + C = X where C is a f- submodule in X, again At is a 

supplement of Ct in Xt ∀  t ∈ (0, 1] therefore   A is f- supplement of C in X. Thus X is amply f- 

supplemented.    

    4.5 Remark 

      Each a direct summand of f- Module is   f- supplement submodule of X. 

Proof 

  Let A be f- submodule of X, then there exist B of X such that (A ⨁ B)t= Xt then (A+B )t= Xt by [ 

6] , therefore A ⨁ B = X then A + B = X and B∩ A  is a proper f- submodule of A. To prove that A 

is supplement of B in X .Suppose that there exist C be a f- submodule in A such that C+B= X. Then 

A= X∩ A= (C+B) ∩ A but ((C+B) ∩ A)t = (Ct+ Bt) ∩ At  ∀  t ∈ (0, 1]  implies that C+ ( B∩ A)  by 

[14] .But by our assumption (B∩ A)  is a proper  f- submodule in A implies that A=C . Thus A is 

supplement of B in X . 

4.6 Proposition 

     Every Loc- hollow fuzzy module is an amply supplemented (supplement fuzzy) is a Ƒuzzy 

submodule. 

 

 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 
 

93 

Proof: 

          Suppose that  𝜒  is a Loc- hollow f- module,    is a unique maximal Ƒuzzy submodule  of 𝜒. 

Since 𝜒   is   Loc- hollow f- module, then we have At+ Xt=)X)t,  t∈ (0,1]  .This leads A+X=X and 

A∩X= A is a small fuzzy  submodule of X by [9]. Therefore X is an amply supplemented module. 

 𝜍onverse of 𝜌roposition (4.6) is not satisfied by the following  

4.7 Example 

    Let M =Z6 , Define 𝜒: M → [0, 1]  and define by: 

A:  M → [0, 1], define by   

A (t) = {
𝑡    𝑖𝑓    𝑥 ∈ 𝑁       

0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 
        ∀  t ∈ (0, 1]   , N =2Z    

And let   B:  M → [0, 1], define by   

B (t) = {
𝑡    𝑖𝑓    𝑥 ∈ 𝐿     

0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 
        ∀  t ∈ (0, 1], L=3Z      

  Clear that At = ⟨2̅⟩ , Bt = ⟨3̅⟩  are two submodule of X and Xt = M = Z6. Hence (A+B)t = (X)t   direct 

summand of Xt is fuzzy submodule of X by[3 ] and there exists a supplement xt of A in Xt such that 

xt ⊆Bt , ∀  t ∈ (0, 1]. Then X =Z6 amply supplemented module. But Z6   is not Loc- hollow y module. 

Since Z6 has no unique maximal submodule. Thus X is supplemented fuzzy module but not Loc- 

hollow fuzzy module 

4.8 Proposition  

        Every Loc- hollow f- module is an indecomposable f- module. 

Proof: 

           Suppose that 𝜒 be a Loc- hollow f- module. ∃ a unique maximal fuzzy submodule say A which 

contains all fuzzy small submodule of X  , let X is decomposable , Then There exists  C,B are  a 

proper fuzzy   submodules of  X  and A,B≠ 01 , such that C ,  𝑩 are  Ƒ- subms of A and 𝜒= C ⊕ 𝑩  

hence (X)t= Ct  ⊕  Bt , ∀  t ∈ (0, 1],But X  is hollow  then either  

B is a small fuzzy of X with B is f- subm of A implies that X=C or C is a small fuzzy of X with C is 

f- submodule of A implies that X=B, which contradiction. Then X is indecomposable module. 

4.9 Remark:  

      The convers of proposition (4.8) it is not always true, as well as, in the next example. 

4.10   Example 

Let   M = Z –module Z, and A = 5Z   

 X: M → [0, 1]   s.t X (x) = {
𝑡    𝑖𝑓    𝑥 ∈   M     

0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 
   ,   ∀  t ∈ (0, 1]. 

clear that Xt =M  and M   is indecomposable module but Loc- hollow module[ 5] . Therefore X is 

indecomposable module but  not Loc- hollow fuzzy module. 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 
 

94 

        Recall that if X finitely generated f- M odule then X/C is finitely generated f- module for every 

C a f- subm X. But the converse is not true. The following proposition shows if X is a hollow f- 

module and X/C a finitely generated f- module. Then X also a finitely generated f- module. 

4.11 Proposition 

        Every Loc- hollow f- module is lifting f- module. 

𝝆roof: 

       Let X   be a Loc- hollow fuzzy module.  ∃   a unique maximal fuzzy submodule says A which 

contains all fuzzy small submodule of X. So X= X ⊕ {01} where {01} is a fuzzy submodule of A, 𝑨  

∩  𝜒= 𝑨 . But X Loc- hollow f- modules. Then   𝑨  𝜒 = 𝑨 is a Ƒuzzy small submodule by [8]   implies 

that X is lifting a fuzzy module. 

4.12 Remark: 

        The convers of proposition (4.11) it is not satisfy we can show that by the following  

4.13 Example:  

      Let X =Z10   , define X: M → [0, 1] define by: 

𝑨:  M → [0, 1], s.t 

A (t) = {
𝑡    𝑖𝑓    𝑥 ∈ 𝑁       
0.5 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 

        ∀  t ∈ (0, 1]    , N= ⟨0̅⟩     

And let   B:  M → [0, 1], define by   

B (t) = {
𝑡    𝑖𝑓    𝑥 ∈ 𝐿        
0.3  𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 

        ∀  t ∈ (0, 1]      , L= 2Z  

C (t) = {
𝑡    𝑖𝑓    𝑥 ∈ 𝐾        
0.25  𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 

        ∀  t ∈ (0, 1]      , K= 5Z  

clear that Xt = M = Z10    is fuzzy module, At =  𝑁 , Bt = L and Ct =  𝐾 are  submodules in Xt .Hence( 

A ⨁ X)t=(X)t   is only direct summand of Xt is fuzzy submodule of Xt, and Xt ∩ At = 01   is  a small 

submodule in Bt = ⟨2̅⟩ . But   Z10 has no unique maximal fuzzy submodule. 

The convers of proposition (4.12) is achieved if the following condition is hold  

4.13 Proposition:  
       Let X be a cyclic fuzzy indecomposable module of an R-module M .If X is lifting fuzzy module, 

then X is Loc- hollow fuzzy module. 

Proof: 

       Let A be a proper f- submodule of X, but 𝜒 is lifting f- module. Then 𝜒 = 𝑨+B, where B is f- 

submodule and A ∩ B is a small f- submodule of A. But X is an indecomposable fuzzy module, 

implies that   B=01 and hence At = Xt, ∀  t ∈ (0, 1] by [9], since Xt is a hollow fuzzy module and hence 

A= X. Which implies that A  ∩  X = A, clear that we have At  ∩ Xt= At, ∀  t ∈ (0, 1]. Then A is a 

small f- submodule of X. So X is hollow fuzzy module and since X is cyclic   fuzzy module. Then X 

is Loc- hollow fuzzy module through Proposition (3.1). 

 

  



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 
 

95 

4.15 Proposition  

      Let C be a maximal f- submodule of a fuzzy module X. If B is a f- supplement of C in X, therefore 

B is a Loc- hollow fuzzy module. 

Proof:  

      Let B is f- supplement of C, let B1 be a proper f- subm of B with B1+B2= B for some f-submodule 

B2 of B and (B1+B2)t= (B)t, ∀  t ∈ (0, 1]. Let C+B = X = C+ B1+B2= X and   B1 is a f- submodule of 

C,  B1= X , by  [1],also  C is a maximal f- subm of X  therefore B1=B, which contradiction with our 

assumption .Thus C +B=X  and since C is a maximal f- submodule of X  so we have  B2=B  implies 

that  C is a hollow fuzzy module. To prove that    C is a cyclic fuzzy module, let xt ⊆X   and   xt ⊆  

C, so there exists (xt ) a small submodule of Xt,  ∀ t∈ (0,1].  Rx +C =X and this implies that Rx= C, 

where (Rx)t = Ct  ∀ t∈ (0,1]  through dependent on minimality of C and by Proposition (3.1). Thus B 

is Loc- hollow fuzzy module. 

 

. Conclusion: 5 

     In this work, we introduced a concept of a hollow f- module, where a module is said to be a hollow 

fuzzy when every subm is a small f-subm. Also, fuzzify these concepts L-hollow module to Loc-

hollow f- modules. Moreover, we generalized numerous properties of Loc-hollow f- modules and got 

several results and fundamental properties of L-hollow f- modules which are argued. Includes the 

relation between hollow f- modules and Loc-hollow f-modules. However, allocated the relation 

between Loc-hollow f- modules and different modules like amply supplemented f- modules, 

indecomposable f- modules, and lifting f- modules. Finally, it is significant to remark that the 

characterizations and properties that are fundamentally related to these concepts are introduced. 

 

References 

1. Zadehi,L.A .Fuzzy sets Information and control. 1965, 8, 333-353.  

2. Zadehi ,M. M. On L- Fuzzy Residual Quotient Modules and P. Primary Submodules, Fuzzy  

    Sets and Systems. 1992,5, 331-344. 

3. Zadehi, M.M. A characterization of L- Fuzzy ideals, F-sets and Systems. 1991, 44,147 -160. 

4. Mashinchi, M. ;Zadehi, M. M. On L-Fuzzy primary Submodule , F- sets and System, 1995,49, 231-

236 

5. Martinez, L. Fuzzy module over Fuzzy rings in connection with Fuzzy ideals of Rings, Fuzzy- sets 

and system. 1996,4 ,843-857. 

6. Rabi, H. J. Prime Fuzzy Submodules and Prime Fuzzy modules.2001, M.Sc. thesis,University of 

Baghdad.  

7.  Mayson, A.H ; Hatam, Y. Khalaf.  Fuzzy regular F- modules. 2002.  M.Sc Thesis of University  

Baghdad. 

8.  Buthyna, N.SH. ; Hatam,Y.KH. Fuzzy-Small submodules with study of Most Important Relevant 

Results. (2020),2nd Al- Noor International Conference for Science and Technology, August 28-29, 

Baku, Azerbaijan, 137-140. 

9. Thear Z.KH ; Nada, K, A, Some Generalization of L –Hollow Modules. 2019. M.ScThesis 

University of Tikrit. 

10. Hassan, K. Marhon ;Hatam ,Y. Khalaf, . Fuzzy closed Submodule and Fuzzy W-closed 

Submodules with of their Generalization. 2020. Ph.D. Thesis of Baghdad  

11. Yahya, T.A. Darzi, On fuzzy supplement submodules.  February. Annals of uzzyMathematics 

and Informatics, 2015, 9, 2, 205-213. 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 53 (2)2022 
 

96 

12. Kumar, R, S. K. Bhambir, Kumar P.  Fuzzy submodule of Some Analogous and Deviation,  

      F- Sets and system. 1995, 70, 125-130. 

13.  Inaam ,m. Hadi, myson, A, Hamill. Cancellation and Weakly Cancellation FuzzyModules. 

Basrah Research's (sciences) , 2011,37 ,4.D. 

14. Hamel, M.A. and Khalaf, H.Y., Fuzzy Semimaximal Submodules. Ibn AL-Haitham Journal For 

Pure and Applied Sciences, 2020,33(4), 137-147.