IHJPAS. 36 (3) 2023 

372 

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

 

   

 

 

 

 

 

 

 

 

 

 

 
*Corresponding Author: Sajidak.mohammed@uokufa.edu.iq 

 

 

 

 

 

Abstract 

In previous our research, the concepts of visible submodules and fully visible modules were 

introduced, and then these two concepts were fuzzified to fuzzy visible submodules and fully 

fuzzy. The main goal of this paper is to study the relationships between fully fuzzy visible modules 

and some types of fuzzy modules such as semiprime, prime, quasi, divisible, F-regular, quasi 

injective, and duo fuzzy modules, where under certain conditions it has been proven that each fully 

fuzzy visible module is fuzzy duo. In addition, there are many various properties and important 

results obtained through this research, which have been illustrated. Also, fuzzy Artinian modules 

and fuzzy fully stable modules have been introduced, and we study the relationships between these 

kinds of modules and fully fuzzy visible modules. Many other intersecting results we found. 

  

Keywords: Fully fuzzy visible modules , fully fuzzy stable modules, fuzzy Artiain modules, fuzzy 

regular rings, fuzzy quasi injective modules. 

 

1. Introduction. 

The notion of fuzzy set has been presented by Zadeh in 1965[1].The fuzzy groups   have been 

introduced by Rosenfeld in 1971 [2]. Then, many other researchers studied different applications 

of fuzzy sets of algebra. The concept of fuzzy modules was presented by Negoita and Relescn in 

1975 [3]. The notion of fuzzy visible submodules was introduced by Sajda K.M. and Buthyna N. 

S. 2021 [4]. The concept of fully fuzzy visible modules has been introduced by Sajda K.M. and 

Buthyna N.S. in 2022 [5]. In this paper, the relationships between fully fuzzy visible modules and 

other different modules have been explained, like fuzzy (prime, semiprime, semisimple, and F-

egular). Also, the notions of stable submodules, fully stable modules, and quasi-injective modules 

doi.org/10.30526/36.3.3092 

Article history: Received 30 October 2022, Accepted 19 December 2022, Published in  July 2023. 

 

 

 

Ibn Al-Haitham Journal for Pure and Applied Sciences 

Journal homepage: jih.uobaghdad.edu.iq 

 

Fully Fuzzy Visible Modules with other Related Concepts 
 

*Sajda K.Mohammed         

Department of Mathematics Faculty of Education 

for Girls, University of Kufa, Iraq 
                                              

Department of Mathematics, College of Education 

for Pure Science Ibn Al-Haitham, University of 

Baghdad, Baghdad, Iraq. 

 

Sajidak.mohammed@uokufa.edu.iq 

 

        Buthyna  N. Shihab  

Department of Mathematics, College of Education 

for Pure Science Ibn Al-Haitham, University of 

Baghdad, Baghdad, Iraq. 

 

bothaina.n.s@ihcoedu.uobaghda.edu.iq 

https://creativecommons.org/licenses/by/4.0/
mailto:Sajidak.mohammed@uokufa.edu.iq
http://en.uobaghdad.edu.iq/?page_id=15060
http://en.uobaghdad.edu.iq/?page_id=15060
mailto:Sajidak.mohammed@uokufa.edu.iq
mailto:Sajidak.mohammed@uokufa.edu.iq
http://en.uobaghdad.edu.iq/?page_id=15060
http://en.uobaghdad.edu.iq/?page_id=15060
mailto:bothaina.n.s@ihcoedu.uobaghda.edu.iq


IHJPAS. 36 (3) 2023 

373 
 

are fuzzyified. Many important properties and results between these above concepts and fully 

fuzzy visible modules have been proven. Through this paper will be a unitary module over a 

commutative ring with identity and for each fuzzy ideal Ʉ of 𝔽, Ʉ (0)=1.  

 

2. Preliminaries. This section contains some definitions and properties that will be needed  in 

our work.  

 

2.1 Definition: Let 𝔽 ≠ ∅  be a set and Ɨ = [0, 1] of the real line (real numbers).  Let ɦ: 𝔽 → Ɨ is a 

function. Then, ɦ  is known as a fuzzy set in 𝔽  (a fuzzy subset of, [1]. Let FSF(𝔽) =

{ɦ: ɦ: 𝔽 → Ɨ is a function }.   

 

2.2 Definition. Let 𝕐 ≠ ∅   and ℭ ∈ 𝐹𝑈𝑆(𝕐) . A level set of 𝕐 with respect to  ℭ  denoted by the 

set ℭ𝑡 , where   ℭ𝑡 ={x∈ 𝕐 ∶  ℭ (x)≥ t} ∀𝑡 ∈ [0,1], [ 6- 8] .   

 

2.3 Definition: Suppose that 𝕄 be an 𝔽 –module and ₮ ∈ 𝐹𝑆𝐸(𝕄) . ℙ is known as a fuzzy 

module of an 𝔽  -module  if, 

1- ₮ (ʑ - ƫ)≥min { ₮ (ʑ), ₮ (ƫ) }, for all ʑ ,ƫ∈  𝕄, 

2- ₮ (rʑ)≥ ₮ (ʑ), for all ʑ∈  𝕄,  r∈  𝔽,  

3- ₮ (0)=1 [9-10] .   

  Let 𝐹𝑈𝑀(𝕄 ) = {₮: ₮ is 𝑎 fuzzy modules of an 𝔽  − module 𝕄 }. 

 2.4 Definition: Let ℭ , ℙ ∈ FUM(𝕄). ℭ is known as a fuzzy submodule of ℙ  if  ℭ ⊆ ℙ [11- 13]  

. 

Let  FUS(ℙ) = {ℭ: ℭ ∈ FUM(𝕄 ) and ℭ ⊆  ℙ }.  

2.5 Definition: Let 𝔽 be a ring and Ί ∈ 𝐹𝑆𝐸(𝔽).  Ί is referred to as a fuzzy ideal of 𝔽, if  ∀  ʑ, ƫ∈

 𝔽  :   

      1-Ί (ʑ -y)≥ min {Ί (ʑ), Ί (ƫ)}  

    2-Ί (ʑ.y)≥ max {Ί (ʑ), Ί (ƫ)}, [14]  . 

 FUI(ℝ ) will be used to represent the set of all fuzzy ideals of 𝔽 . 

 

 2.6 Definition: Let ℙ ∈ 𝐹𝑈𝑀(𝕄),ℭ ∈ 𝐹𝑈𝑆(ℙ) and Ί ∈ 𝐹𝑈𝐼(𝔽).The product (Ί ℭ)(x)  = 

{
𝑠𝑢𝑝 {𝑖𝑛𝑓 {Ί (𝑟1), … , Ί (𝑟𝑛), ℭ(𝑥1), … , ℭ(𝑥𝑛)𝑓𝑜𝑟 𝑠𝑜𝑚𝑒 𝑟𝑖  ∈ 𝔽, 𝑥𝑖 ∈ 𝕄, 𝑛 ∈ 𝑁,
 0                                                                                            𝑜. 𝑤.                               

𝑥 = ∑ 𝑟𝑖 𝑥𝑖
𝑛
𝑖=1       

,[9]  .          

Note that Ί ℭ ∈ 𝐹𝑈𝑆(ℙ) if  Ί (0)=1,  [9]. And (Ί ℭ)𝑡 = Ί 𝑡 ℭ𝑡for each 𝑡 ∈ [0,1],  [15].  

 

2.7 Definition Let ᵮ be a mapping from a set 𝕄 into a set N, Let 𝔸 ∈ FSE(𝕄) and 𝔹 ∈ 𝐹𝑆𝐸(𝑁). 

The image of 𝔸 denoted by ᵮ (𝔸) ∈ 𝐹𝑆𝐸(𝑁) defined by :  

ᵮ(𝔸)(ℯ) = {
{𝑠𝑢𝑝 {𝔸(𝑧): 𝑧 ∈ ᵮ−1(ℯ), 𝑖𝑓 ᵮ−1(ℯ) ≠ ∅, 𝑓𝑜𝑟 𝑎𝑙𝑙 ℯ ∈ 𝑁

0                                                  𝑜. 𝑤.                              
 , 

and the inverse image of 𝔹 denoted by 𝑓 −1(𝔹) ∈ 𝐹𝑆𝐸(𝕄)𝑖𝑠 defined by:  

ᵮ−1(𝔹)(𝑥) = 𝔹(ᵮ(𝑥)), for all ∈ 𝕄 [15-16] . 

 

2.8 Definition: Let 𝕏 ∈FUM(𝕄1) and 𝕐 ∈FM(𝕄2). ℊ ∶ 𝕏 → 𝕐  is called a fuzzy homomorphism 

if ℊ: 𝕄1 → 𝕄2 is 𝔽-homomorphism and 𝕐(ℊ(𝒽) = 𝕏(𝒽)     for each 𝒽 ∈ 𝕄1 [15] .  

 



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2.9 Definition: Let ∅ ≠  ℭ ∈ 𝐹𝑈𝑆(ℙ). The fuzzy annihilator of A denoted by F-ann ℭ is defined 

by: (F-ann ℭ)(r)=sup{t∶ t ∈[0,1],𝑟𝑡  ℭ ⊆ 01  },for all r∈ 𝔽 [17-18]  .  

   Note that that   F-ann ℭ =(01∶ ℭ) ," [16]. "Hence ((𝐹 − 𝑎𝑛𝑛 ℭ))𝑡  ⊆  ann ℭ𝑡  [19].    

 

2.10 Remark: For A ∈ FUM(𝕄).We let ℭ∗ = {𝑥 ∈ 𝕄: ℭ(𝑥) = 1}[20 − 21] . 

 

2.11 Proposition: Let ℙ ∈ FUM(M) .Then  𝐹 − 𝑎𝑛𝑛 ℙ ∈ 𝐹𝑈𝐼(𝔽)[9] . 

 

2.12 Definition: Let Ҡ ∈FUI(𝔽). Ҡ is called  a principle fuzzy ideal if  ∃𝓀𝑡 ⊆ Ҡ such that 

 Ҡ = (𝓀𝑡 ) for each 𝒾𝑠 ⊆ Ҡ , ∃ a fuzzy singleton ℯ𝑙 of 𝔽 such that 𝒾𝑠 = ℯ𝑙 𝒽𝑡 , where 𝑠, 𝑙, 𝑡 ∈ [0,1] 

, which is  Ҡ = (𝓀𝑡 ) = {𝒾𝑠 ⊆ 𝐻: 𝒾𝑠 = ℯ𝑙  𝓀𝑡 for some fuzzy singleton 𝑎𝑙 of 𝔽} [22] .  

 

 2.13 Definition: Let Ϯ ∈ 𝐹𝑈𝑀(𝕄)  and Ϯ≠ Đ∈FUS (Ϯ).Then Đ is known to as a fuzzy visible 

submodule if  Đ=ƗĐ ∀  non-empty  Ɨ ∈ 𝐹𝑈𝐼(𝔽).  ɬ∈ 𝐹𝑈𝐼(𝔽)  is defined visible if it is visible of a 

fuzzy  𝔽 -module 𝔽 [4]. 

 

2.14 Definition: Let 𝕐 ∈ 𝐹𝑈𝑀(𝕄). Then 𝕐 is said to be fully fuzzy visible module if for 

any 𝕐 ≠ ℨ ∈ 𝐹𝑈𝑆 (𝕐)  is a fuzzy visible [5].  

 

2.15 Proposition: Let 𝕐 ∈ 𝐹𝑈𝑀(𝕄) . Then ℙ is fully fuzzy visible module if and only if 𝕐t is 

fully visible module,∀ 𝑡 ∈ (0,1] [5] . 

 

2.16 Definition: Let 𝕏 ∈ 𝐹𝑈𝑀(𝕄).  𝑋≠ ℭ∈FUS(𝕏), ℭ is termed as a semiprime fuzzy submodule 

if for each fuzzy singleton 𝑟𝑙   of 𝔽, 𝑥𝑠 ⊆ 𝕏, 𝑟𝑙
2𝑥𝑠 ⊆ ℭ implies 𝑟𝑙 𝑥𝑠 ⊆ ℭ [23] . 

 

2.17 Remark: We assume that if   ℭ∗ = 𝔔∗ . Then , ℭ = 𝔔 is called Condition ( *) [20] . 

 

3. Fully Fuzzy Visible Modules with Different Modules: In this section, the relationships 

between fully fuzzy visible modules and other fuzzy modules such as (semiprime, prime quasi 

prime, fully stable, Artiain, essential and quasi injective) have been investigated. Many interesting 

outcomes were identified. 

3.1 proposition: Let  ℙ ∈ FUM(𝕄) over principle fuzzy ideal ring. Then ℙ is fully fuzzy visible 

if and only if every proper fuzzy submodule of  ℙ is fuzzy semiprime.  

Proof: Let ℙ be a fully fuzzy visible, then ℙt is a fully visible, where 𝑡 ∈ (0,1]  by [5,proposition 

3.2] and let ℙ ≠ 𝔚 ∈ 𝐹𝑈𝑆(ℙ)  . Then 𝔚t is a proper submodule of ℙt and hence 𝔚t is a 

semprime submodule by [ 24 , proposition 2.6.1 ] . Therefore 𝔚 is a fuzzy semiprime by [23 , 

proposition 3.2.6] . Conversely, let  ℙ ≠ 𝔚 ∈ 𝐹𝑈𝑆(ℙ)  , then 𝔚 is a fuzzy semprime and hence 

𝔚t is semiprime by [23, proposition 3.2.6], therefore ℙt is a fully visible by [    24 , proposition 

2.6.1] and hence ℙ is fuzzy fully visible. 

3.2 Proposition: Let  ℙ ∈ FUM(𝕄) over principle fuzzy ideal ring   and   ℙ ≠ 𝔚 ∈ 𝐹𝑈𝑆(ℙ). 

Then, 𝔚 visible if and only if 𝔚 divisible. 

Proof: Assume that 𝔚  is visible then ∀ 𝑟𝑙 ≠ 01, we have < 𝑟𝑙 > 𝔚 = 𝔚 , then 𝑟𝑙 𝔚 =

𝔚.Therefore 𝔚 is divisible. Conversely, clear. 

3.3 Proposition: Let 𝕐 be a fully fuzzy visible module over 𝔽. Then every proper fuzzy 

submodule of 𝕐 is divisible.  



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Proof: Let 𝕐 be a fully fuzzy visible module, then every proper fuzzy submodule of 𝕐 is visible 

and hence is divisible by above proposition.  

3.4 Proposition: If ℙ is a fully fuzzy visible module, then ℙ is F – regular fuzzy module. 

Proof: By using definition( 2.14) and [4, proposition 3.17], the result is true. The converse of the 

above proposition is not correct as we see in the following example. 

 Consider 𝕄 = 𝑍6  and 𝔽 = 𝑍. Let ℙ: 𝑍6 → [0,1], as ℙ(𝑥) = 1 ∀𝑥 ∈ 𝑍6. 𝑍6 𝑎𝑠 𝑍 − module   is F-

regular by [24, p97 ] and hence ℙ is F-regular by [25 , proposition 3.1.3]. But  𝑍6  is not fully 

visible by [24, p97] and hence ℙ is not fully visible  . 

3.5 Proposition: Let ℙ be a fuzzy divisible module over a field, then ℙ is a fully fuzzy visible 

module. 

Proof: From [25, proposition 3.1.8], we have ℙ is an F- regular fuzzy module, then ℙt is a F- 

regular module, for every 𝑡 ∈ (0,1] by [25, proposition 3.1.2]. ℙt is divisible because ℙ is a fuzzy 

divisible module by [ 23, proposition 2.2.12] and hence every proper pure submodule of ℙt is 

visible by [26 , proposition 1.11]. Therefore ℙt is fully visible, then ℙ is fuzzy fully visible. 

3.6 Corollary: Suppose that  ℙ is a non-constant fully fuzzy visible module over a fuzzy integral 

domain 𝔽. Then, ℝ is a fuzzy field. 

Proof: Depending on proposition (3.4) and [ 22, remark and examples 1.3.4(4)    ], we get the 

result. 

3.7 Corollary: Let ℙ ∈ FUM(𝕄). If ℙ is fully visible, then 𝐹 − 𝐽(ℙ) = 01, where ℙ satisfies 

condition(∗). 

Proof:   ℙ∗ is a fully visible because ℙ is a fully fuzzy visible by [5, proposition3.2] and 

hence𝐽(ℙ∗) = {0} = (01)∗, by [ 24, corollary 2.6.6]. But 𝐽(ℙ∗) = (𝐹 − 𝐽(ℙ))∗
= (01)∗, then 𝐹 −

𝐽(ℙ) = 01.  

3.8 Definition: Let ℙ ∈ FUM(𝕄). Then ℙ is called fuzzy Artiain module if every descending 

chain fuzzy submodules of ℙ is finite.  

3.9 Theorem: If ℙ ∈ 𝐹𝑈𝑀(𝕄) Artiain 𝔽 -module, then ℙ∗ is Artiain modules. 

Proof: Let ℙ ∈ 𝐹𝑀(𝕄) Artiain ℝ -module and  𝐴1 ⊇ 𝐴2 ⊇ ⋯ be descending chain of 

submodules of ℙ∗. For each positive integer i, we define the mapping ℙ𝑖 : 𝕄 → [0,1] by ℙ𝑖 (𝑥) =

{
1 𝑖𝑓 𝑥 ∈ 𝐴𝑖
0 𝑜. 𝑤

 , clearly, ℙ𝑖 ∈ FS(ℙ) . Then ℙ1 ⊇ ℙ2 ⊇ ⋯ is descending chain of fuzzy 

submodules of ℙ, so there exist a positive integer m such that ℙ𝑚 = ℙ𝑚+ℎ   for every positive 

integer h. It is clear that (ℙ𝑖 )∗ = 𝐴𝑖   . Now, let h be a positive integer and 𝑎 ∈ 𝐴𝑚+ℎ so, ℙ𝑚(𝑎) =

ℙ𝑚+ℎ (𝑎) = 1, hence 𝑎 ∈ 𝐴𝑚. Thus 𝐴𝑚 ⊇ 𝐴𝑚+ℎ ⊇ 𝐴𝑚. Therefore ℙ∗ is Artiain. The converse of 

above theorem is not true as in the following example. 

Let 𝕄 = 𝔽 and 𝔽 =  ℝ . Express  ℙ: 𝕄 → [0,1] by ℙ(x) = 1 ∀x ∈ 𝕄.  

Clearly, ℙ ∈ FUM(𝕄). Describe ℙ: 𝕄 → [0,1] by  

ℙ𝑚 (𝑥) = {
1 𝑖𝑓 𝑥 = 0
1

2𝑚
𝑖𝑓 𝑥 ≠ 0

  , 𝑚 ∈ 𝑍+. Clear that ℙ𝑚 ∈ 𝐹𝑈𝑆(ℙ)  for each m. Then ℙ1 ⊇ ℙ2 ⊇ ⋯ 

is an infinite strictily descending chain of fuzzy submodules of ℙ. Thus ℙ is not fuzzy Artiain. 

But, ℙ∗ =  𝕄, is Artiain 𝔽 −module.  

3.10 Corollary: Let ℙ be a fully fuzzy visible Artiain module on 𝔽. Then ℙ is fuzzy semisimple 

module provided that ℙ satisfies condition (∗).  



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Proof: ℙ∗ is a fully visible module since ℙ is fully fuzzy visible and ℙ∗ is Artiain by theorem 3.9. 

Therefore ℙ∗ is semisimple by [24, proposition 2.6.7]. Then, ℙ is fuzzy semisimple by [20, 

proposition 2.8(1)]. 

3.11 Definition: A fuzzy ring 𝔽 is called regular if and only if every fuzzy singleton in 𝔽 is fuzzy 

regular, i.e. ∀ fuzzy singleton 𝑥𝑡 of 𝔽, 𝑡 ∈ (0,1], there exist a fuzzy singleton 𝑦𝑘 of 𝔽 such that 

𝑥𝑡 = 𝑥𝑡 𝑦𝑘 𝑥𝑡 .  

3.12 Proposition: A fuzzy ring 𝔽 is regular if and only if 𝔽 is regular ring. 

Proof: Suppose that 𝔽 is regular, then ∀𝑥 ∈ 𝔽, ∃𝑦 ∈ 𝔽, s.t. 𝑥 = 𝑥𝑦𝑥 , hence  

𝑥𝑡 = (𝑥𝑦𝑥)𝑡 = 𝑥𝑡 𝑦𝑡 𝑥𝑡 , where 𝑥𝑡 , 𝑦𝑡  fuzzy singleton of 𝔽, 𝑡 ∈ (0,1]. Therefore, a fuzzy ring 𝔽 is 

regular. Conversely, let 𝑀𝔽 be a fuzzy ring over 𝔽 , then ∀ fuzzy singleton xt of 𝔽, ∃𝑦𝑘 fuzzy 

singleton of 𝔽 such that 𝑥𝑡 = 𝑥𝑡 𝑦𝑘 𝑥𝑡 = (𝑥𝑦𝑥)𝑚𝑖𝑛{𝑡,𝑘}, hence 𝑥 = 𝑥𝑦𝑥 and 𝑡 = 𝑚𝑖𝑛{𝑡, 𝑘}. 

Therefore 𝔽 is a regular. 

3.13 Corollary: Let ℙ be a fully fuzzy visible and Artiain module on 𝔽, satisfied condition(*), 

then 𝑀𝔽/(𝐹 − 𝑎𝑛𝑛𝑥𝑡 )∗ is fuzzy regular ∀𝑥𝑡 ⊆ ℙ.  

Proof: ℙ∗ is fully visible and Artiain module as in corollary (3.10). Then 𝔽/(𝐹 − 𝑎𝑛𝑛𝑥𝑡 )∗ is 

regular by [24, corollary 2.6.10], then M𝔽/(F − 𝑎𝑛𝑛𝑥𝑡 )∗ is fuzzy regular by [27, theorem 3.2.10] 

and proposition (3.12) . 

3.14 Proposition: Let ℙ ∈ 𝐹𝑈𝑀(𝑀) over principle fuzzy ideal ring. If every ℙ ≠ 𝔚 ∈

𝐹𝑈𝑆(ℙ) over 𝔽 is divisible, then ℙ is fuzzy fully visible and hence ℙ is F- regular fuzzy.  

Proof: By using proposition 3.2, we have ℙ  a is fuzzy fully visible, then ℙ is F-regular by 

proposition (3. 4). 

3.15 Proposition: Let 𝔽 be a principle fuzzy ideal ring and ℙ is a fuzzy divisible, then the 

following is equivalent: 

1. ℙ is a fully fuzzy visible module.  

2. Each ℙ ≠ ᵹ ∈ 𝐹𝑈𝑆(ℙ) is pure.  

3. Each ℙ ≠ ᵹ ∈ 𝐹𝑈𝑆(ℙ)  is divisible.  

4. Each ℙ ≠ ᵹ ∈ 𝐹𝑈𝑆(ℙ) is visible. 

Proof:1 ⟹ 2 directly from [4, proposition 3.17].  

2 ⟹ 3 Since 𝔽 be a principle fuzzy ideal ring and ℙ is a fuzzy divisible, then by[4, proposition 

3.19] we have each proper submodule of ℙ is visible and hence divisible. 

3 ⟹ 4 Directly from proposition 3.2. 

4 ⟹ 1 Clear.  

3.16 Proposition: Let ℙ be a fully fuzzy visible module over a local ring, then ℙ is a fuzzy 

semisimple module such that ℙ  satisfied condition (∗). 

Proof: ℙ∗ is a fully visible module by [5, proposition 3.2], then ℙ∗ is semisimple by [24   , 

proposition 2.6.13], hence ℙ is a semisimple by [20, proposition. 2.8(1)]. 

3.17 Proposition: Let ℙ be a fuzzy divisible over principle fuzzy ideal ring, then ℙ is fully visible. 

Proof: Depending on [4, proposition 3.19], every proper fuzzy submodule of ℙ is visible and 

hence ℙ is fully visible. 

3.18 Proposition: Let ℙ be a fully fuzzy visible module over 𝔽, then ℙ is a fuzzy torsion free. 

Proof:  ℙt is fully visible ∀ 𝑡 ∈ (0,1] by [5, proposition 3.2], then ℙt torsion free by [ 24 

,proposition 2.6.16]. Therefore ℙ is fuzzy torsion free by {17, proposition 2.2.23]. 

 



IHJPAS. 36 (3) 2023 

377 
 

3.19 Proposition: Let ℙ be a fully fuzzy visible module, ℙ ≠ 𝔚 ∈ 𝐹𝑈𝑆(ℙ). Then, 𝔚 is a prime 

submodule. 

Proof: Because ℙ is a fully fuzzy visible module and ℙ ≠ 𝔚 ∈ 𝐹𝑈𝑆(ℙ),   then 𝔚 is pure, hence 

𝔚 is weakly pure fuzzy submodule of ℙ. Therefore 𝔚 is a prime fuzzy by [17, proposition 2.2.22]. 

3.20 Corollary: If ℙ is a fully fuzzy visible module, then every proper fuzzy submodule 𝔚 of ℙ 

is quasi prime.  

Proof: Depending proposition 3.19 and [23, proposition 3.2.2], the result hold. 

3.21 Corollary: Every fully fuzzy visible module is fuzzy T-regular. 

Proof: Depending on proposition (3.4) and [22, remark and examples 2.1.5(1)]. 

The converse of the corollary (3.21) is not true as we see in the following example: 

Let 𝕄 = 𝑍4, 𝔽 = 𝑍. Define 𝑃: 𝕄 → 𝑍4 as ℙ(𝑥) = {
1 if 𝑥 ∈ 𝑍4
0 𝑜. 𝑤

, clearly,  ℙ ∈ FUM(𝕄), then 

ℙt = Z4  , ∀ 𝑡 ∈ (0,1]  as Z-module is T-regular by [ 24 , p99   ], then ℙ is fuzzy T-regular by [22, 

proposition 2.1.2]. But Z4 as Z-module is not fully visible module by [24, remark  and examples 

2.1.2], then ℙ is not fully fuzzy visible. 

3.22 Definition: Let ℙ ∈ FUM(𝕄) , 𝔚 ∈FUS(ℙ). We say that   𝔚   is stable if  𝑓(𝔚) ⊆ 𝔚 for 

each fuzzy 𝔽 -homomorphism 𝑓: 𝔚 → ℙ.  

3.23 Proposition: Let ℙ ∈ FUM(𝕄) and 𝔚 ∈ 𝐹𝑈𝑆(ℙ)  constant on 𝑘𝑒𝑟 𝑓. Then 𝔚 is fuzzy stable 

if and only if 𝔚∗ is stable and ℙ satisfies Condition (∗).  

Proof: Let 𝔚 be a fuzzy stable. Then, 𝑓(𝔚) ⊆ 𝔚 by definition (3.21). Since 𝔚 is constant on 

ker 𝑓, then 𝑓(𝔚∗) ⊆ 𝔚∗ by [27, lemma 3.2.5]. Therefore 𝔚∗ is stable. Conversely, let 𝔚∗ is 

stable. Then 𝑓(𝔚∗) ⊆ 𝔚∗ and hence (𝑓(𝔚 ))∗ ⊆ 𝔚∗ by [27, lemma 3.2.5], since 𝔚 satisfies 

condition( ∗), then 𝑓(𝔚) ⊆ 𝔚. Therefore 𝔚 is fuzzy stable. 

3.24Example: Let 𝕄 = 𝔽 and 𝔽 = ℝ. Define Ҡ: 𝔽 → [0,1] by Ҡ(𝑥) = 1 ∀𝑥 ∈ 𝔽 . Consider 

𝔛: 𝔽 → [0,1]  as 𝔛(𝑥) = {
1 𝑖𝑓 𝑥 ∈ Q
0 𝑜. 𝑤

. Clear that 𝔛 ∈ 𝐹𝑈𝑆(Ҡ). Ҡ∗ = ℝ , 𝔛∗ = Q . We know that 

𝑄 is stable of 𝔽, then 𝔛 is a fuzzy stable of Ҡ by proposition (3.22), where 𝔛is coustant on 𝑘𝑒𝑟 𝑓.  

3.25 Example: Let 𝕄 = 𝑍 and 𝔽 = 𝑍. Define ℙ: 𝑍 → [0,1] by ℙ(𝑥) = 1 ∀𝑥 ∈ 𝑍. 

Suppose that 𝔚: 𝑍 → [0,1] defined as 𝔚(𝑥) = {
1 𝑖𝑓 𝑥 ∈ 2𝑍
0 𝑜. 𝑤

. It is clear that 𝔚 ∈ 𝐹𝑈𝑆(ℙ), ℙ∗ =

𝑍 and 𝔚∗ = 2𝑍. Then 𝔚∗ is not stable by [28, example and remarks 1.2(a)], and, hence, 𝔚 is not 

fuzzy stable by proposition 3.23. 

3.26 Definition: Let ℙ ∈ FUM(M). ℙ is termed to be fuzzy fully stable if and only if every    ɮ ∈

𝐹𝑈𝑆(ℙ)  is fuzzy stable.  

3.27 Proposition: Let ℙ be a fuzzy 𝔽 –module, such that every fuzzy submodule of ℙ constant on 

𝑘𝑒𝑟 𝑓 and satisfies condition(*). Then ℙ is a fuzzy fully stable if and only if ℙ∗ is fully stable.  

Proof: In a similar way of proposition (3.23). 

3.28 Proposition: Let ℙ be a fully fuzzy visible 𝔽 -module and Endℝ(ℙ∗) is commutative. Then 

ℙ is fuzzy fully stable such that ∀ 𝔚 ⊆ ℙ constant on 𝑘𝑒𝑟 𝑓 and satisfies condition (*). 

Proof: ℙ∗ is a fully visible by[3, proposition 3.2], then ℙ∗ is fully stable by [24, proposition 

(2.6.22) ], and hence each submodule of ℙ∗ will be stable, then ∀ 𝔚 ∈ 𝐹𝑆(ℙ) , 𝔚∗ will be stable, 

then by proposition 3.22, 𝔚 is a stable fuzzy submodule. Therefore, ℙ is fuzzy fully stable. 

 

 



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3.29 Proposition: Let ℙ be a fully fuzzy visible module over 𝔽. Then each Jℙ ∈ 𝐹𝑈𝑆(ℙ)   is an 

essential submodule of ℙ for each non-emptyJ ∈ 𝐹𝑈𝐼(𝔽). 

Proof: Let ξ be a non-trivial fuzzy submodule of ℙ such that Jℙ ∩ 𝔛 = 01. But 𝔛 is visible and 

hence ξ is pure. Then Jξ = Jℙ ∩ ξ = 01, so Jξ = 01 and hence ξ = 01. Therefore Jℙ is a fuzzy 

essential.  

3.30 Proposition: Let ℙ be a fully fuzzy visible multiplication module, then every non-empty 

fuzzy submodule of ℙ is an essential of ℙ. 

Proof: Because ℙ is fully fuzzy visible, then every fuzzy submodule Jℙ is an essential for each 

non-empty J ∈ 𝐹𝑈𝐼(𝔽) by proposition (3.29). But ℙ is fuzzy multiplication, then every non – 

empty fuzzy submodule of ℙ is essential. 

3.31 Proposition: Let ℙ be a fully fuzzy visible module over a domain 𝔽 and EndR(ℙt) is 

commutative ring. Then ℙ is fuzzy divisible.  

Proof : Since ℙt is fully visible, then ℙt is divisible by [ 24, proposition 2.6.42(2)]. Then ℙ is 

divisible by  [17].  

Let 𝔓 ∈ 𝐹𝑈𝑀(𝕄) and  𝔊 ∈ 𝐹𝑈𝑀(𝑁). If ᵮ ∶ 𝕄 → 𝑁 is a 𝔽 –module homomorphism and 

𝔊(ᵮ(𝑥)) ≥ 𝔓(𝑥), ∀ 𝑥 ∈ 𝕄, then ᵮ is called a fuzzy homomorphism and denoted by ᵮ: 𝔓 → 𝔊 

[29].This definition can be used to introduce the following concept. 

3.32 Definition : Let ℙ ∈ 𝐹𝑈𝑀(𝕄). ℙ is termed to be fuzzy quasi Injective if for each 

submodule N of 𝕄 and each fuzzy homomorphism ᵮ ∶ 𝔚 → ℙ, where 𝔚 ∈ 𝐹𝑈𝑀(𝑁) can be 

extended to fuzzy endomorphism  ᶃ such that ᵮ = ᶃ ∘ i, where i: 𝔚 → 𝕄 inclusion 

homomorphis. 

3.33 Proposition: Let 𝕍 ∈ 𝐹𝑈𝑀(𝕄). ℙ is fuzzy quasi injective iff 𝕄 is quasi injective and 

𝕍(ℯ) = 1 ∀ℯ ∈ 𝕄. 

Proof: Let 𝕍 be a fuzzy quasi injective 𝔽 -module and N be a submodule of 𝕄. 

Define 𝔚: N → [0,1] by 𝔚(𝓆) = {
1 if 𝓆 ∈ N
0 o. w

, then  

∀ ᵮ: 𝔚 → 𝕍 be a fuzzy homomorphism  ∃  ᶃ: 𝕍 → 𝕍 s.t. ᵮ = ᶃ ∘ i and hence ∀ submodule N of 

𝕄 and ᵮ: N → 𝕄, there exist g: 𝕄 → 𝕄 such that ᵮ = ᶃ ∘ i.Now, if 𝕍(𝓆) ≠ 1, since 𝕍(ᶃ(𝓆)) ≥

𝕍(𝓆) by definition of fuzzy homomorphism, then 𝕍(ᶃ(𝓆)) ≥ 𝕍(𝓆) = 𝕍(i(𝓆)) ≥ 𝔚(𝓆) = 1, 

hence 𝕍(𝓆) = 1 ∀𝓆 ∈ 𝕄.  

Conversely, Let 𝕄 be a quasi-injective, N be a submodule of 𝕄 and 𝕍(𝓆) = 1 ∀𝓆 ∈ 𝕄, let 

𝔚 ∈ 𝐹𝑈𝑀(𝑁). Then ∀ ᵮ: N → M  ∃ ᶃ: 𝕄 → 𝕄s.t. ᵮ = ᶃ ∘ i,  i: N → 𝕄  inclusion 

homomorphism, hence ∀ ᵮ ∶ 𝔚 → 𝕍 ∃ ᶃ: 𝕍 → 𝕍 s.t.  ᵮ = ᶃ ∘ i, where (ᶃ ∘ i)(𝓆) =

𝕍(ᶃ ∘ i)(𝓆) = 𝕍(ᶃ(𝓆)) = 𝕍(𝓆) = 1 ≥ 𝕍(𝓆) = 1 ≥ 𝔚(𝓆) . 

3.34 Definition: A fuzzy ring 𝔽 is self injective if it is fuzzy injective 𝔽 -module.  

3.35 Corollary: Suppose that ℙ is a fuzzy fully visible 𝔽 - module 𝕄 and EndR(P∗) is 

commutative, where ℙ satisfies condition (∗). If ℙ is a fuzzy quasi injective module and ℙ (x)=1 

∀ x ∈ 𝕄, then  

1. Every fuzzy submodule 𝔚 of ℙ constant on ker ᵮ  is fuzzy duo module. 

2. Every homomorphism image constant on ker ᵮ is duo module. 

3. A fuzzy ring γ on End𝔽(ℙ∗) is self injective. 

Proof: 1. Since ℙ is fully fuzzy visible and EndR(ℙ∗) is commutative, then ℙ∗ is fully visible.  

But ℙ is fuzzy quasi injective, then 𝕄 is a quasi injective and ℙ(x) = 1 ∀x ∈ 𝕄. Therefore   



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379 
 

ℙ∗is duo by [5, proposition 3.8]. Let 𝔚 ∈ 𝐹𝑈𝑆(ℙ), then 𝔚∗ submodule of ℙ∗, then 𝔚∗ is duo 

which implies that 𝔚 is duo by [5, proposition 3.8 ]. 

2. Since every fuzzy homomorphic image is a fuzzy submodule of  ℙ, then by (1), we get the 

result. 

3. Since End𝔽(ℙ∗) is End𝔽(ℙ∗) −injective by [24, proposition 2.6.25], then γEnd𝔽 (ℙ∗) is a fuzzy 

self injective [29, proposition 4.3(i)]. 

3.36 Corollary: Let ℙ be a finitely generated fully visible module over Dedkined domain, then 

ℙ is fuzzy duo multiplication module, where ℙ satisfies condition (∗) and every fuzzy 

submodule of ℙ constant on ker f. 

Proof : ℙ∗ is finitely generated module, then ℙ∗ is duo multiplication module by [24, corollary 

2.6.26], but ℙ∗ is fully visible, then ℙ is duo multiplication by [25] and [ 5,proposition 3.8 ] . 

3.37 Proposition: If ℙ ∈ 𝐹𝑈𝑀(𝕄)   over a fuzzy principle ideal ring and 𝔚 ∈ 𝐹𝑈𝑀(ℙ) be a   

such that ∀𝑥𝑡 ⊆ 𝔚 and a fuzzy singleton   𝑟ℓ  of 𝔽 , 𝑥𝑡 = 𝑟ℓ𝑠𝑘 𝑥𝑡  for some fuzzy singleton 

𝑠𝑘 of ℝ , then ℙ is a fully visible fuzzy module . 

Proof: 

Let 𝑥𝑡 ⊆ 𝔚 and fuzzy singleton 𝑟ℓ of 𝔽. Then 𝑥𝑡 ⊆< 𝑥𝑡 >⊆ 𝔚, then 𝑥𝑡 = 𝑠𝑘 𝑥𝑡 for some 𝑠𝑘 ⊆

𝑅. Then for each 𝑟ℓ ⊆ 𝔽, we get  𝑠𝑘 𝑥𝑡 = 𝑟ℓ𝑠𝑘 = 𝑟ℓ𝑠𝑘 𝑥𝑡 ⊆ 𝑟ℓ𝔚. Therefore,  𝑥𝑡 ⊆ 𝑟ℓ𝔚, but 𝔽 is a 

fuzzy principle ideal ring, then 𝑥𝑡 ⊆ 𝐽𝔚 and hence 𝔚 ⊆ 𝐽𝔚, but 𝐽𝔚 ⊆ 𝔚. Therefore 𝔚 = 𝐽𝔚  

for each nonempty fuzzy ideal J of 𝔽and hence 𝔚 is a fuzzy visible submodule, thus the result 

holds.  

3.38 Proposition: Let 𝔽 be a principle ideal field and 𝔚 be a fuzzy submodule of a fuzzy 

module of an 𝔽 -module 𝕄. Then  

1. ℙ is a fully fuzzy visible.  

2.γ𝔽/(F − annxt)∗  is fuzzy negular ring ∀xt ⊆ ℙ, where γ𝔽  is a fuzzy ring of 𝔽. 

3.∀ xt ⊆ ℙ ∃rℓ  fuzzy singleton of 𝔽,  xt = rℓhkxt, k ∈ (0,1].  

Proof: 1 ⇒ 2  P∗ is fully visible by propositon (2.15) and hence 𝔽/(F − annxt)∗ is regular ring 

by [24, proposition 2.6.27].Therefore γ𝔽/(F − annxt)∗ is fuzzy negular ring by [27, theorem 

3.2.10] and proposition 3.12.  

2 ⟹ 3 Since γ𝔽/(F − annxt)∗  is fuzzy regular, then 𝔽/(F − annxt)∗ is regular ring by 

[24,theorem 3.2.10],  hence ∀x ∈ ℙ𝑡 , r ∈ 𝔽, x = rhx  for some h ∈ 𝔽 by [  24 ,proposition 

2.6.27]. Therefore,  ∀ xt ⊆ ℙ, ∃γℓ fuzzy singlet of 𝔽 s.t.  xt = rℓhkxt , t = min{ℓ, k, t} . 

 2 ⟹ 3 Directly from proposition 3.37   

3.39 Corollary: If 𝔽 is a principle ideal field, then 𝕄𝔽 is fuzzy regular, if and only if 𝕄𝔽is a  

fuzzy 𝔽 -module is fully visible.  

Proof: Let 𝕄𝔽 be a fuzzy regular ring, then 𝔽 is regular ring by proposition 3.12, then 𝔽 is fully 

visible by [  24, corollary 2.6.28], and hence M𝔽 is fully visible. 

3.40 Corollary: If 𝔽 is a field and ℙ ∈ 𝐹𝑈𝑀(𝕄) such that 𝕄𝔽  is a fuzzy ring on  𝔽, then ℙ is 

fully fuzzy visible module, if 𝕄𝔽/(F − ann ℙ) 𝑡  is a fuzzy regular ring , where  (F − annℙ)t =

annℙt , ∀ 𝑡 ∈ (0,1]. 

Proof: Since 𝕄𝔽/F − annℙ  is  a fuzzy regular, then 𝔽/(F − annℙ)t is a regular ring by 

[27,theorem 3.2.10] and proposition 3.15, but  (F − annℙ)t =  annℙ𝑡,  ∀ 𝑡 ∈ (0,1] , then  ℙ𝑡 is 

fully visible by [24 ]. Therefore ℙ is fully fuzzy visible. 

3.41 Proposition: If ℙ is divisible over P.I.D. and d F-ann ℙ is semimaximal ideal of 𝔽. Then ℙ 

is a fully fuzzy visible, where (F − annℙ)∗ = annℙ∗.  And ℙ(x) = 1 ∀x ∈ 𝕄. 



IHJPAS. 36 (3) 2023 

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Proof: P∗ is divisible module by [23], since (F − annℙ)∗ = annℙ∗ and F − annℙ is 

semimaximal, then annP∗ is semimaximal ideal by [20,  proposition (2.4)]. Then ℙ∗ is fully 

visible [ 24   ,proposition 2.6.38]. Therefore ℙ is fuzzy fully visible.   

3.42 Proposition: Let γ𝔽 be a fuzzy visible ring defined as γ𝔽(r) = 1 ∀r ∈ 𝔽, if ℙ is a fuzzy 𝔽 -

module M, then ℙ is flat.   

Proof: Since (γ𝔽)t = 𝔽, then 𝔽 is visible, then M is flat by [  24, proposition 2.6.39]. Therefore 

ℙ is fuzzy flat by [ 29,  theorem 3.15]   

4. Conclusion: In this paper, a number of characteristics and properties that explain the 

relationship of fully fuzzy visible modules with other modules are proved. In addition  many 

useful examples with a collection of important results are obtained.    

 
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