IHJPAS. 36 (4) 2023 

407 

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

 

   

 

 
 

 

 

 

 

 

 

*Corresponding Author: mbhmsc2015110@gmail.com 

 

 
  

 

 

Abstract 

    Let 𝑅 be a commutative ring with 1 and 𝑀 be left unitary  𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒. In this paper we 

introduced and studied concept of semi-small compressible module (a 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒 𝑀  is said 

to be semi-small compressible module if 𝑀 can be embedded in every nonzero semi-small 

submodule of 𝑀. Equivalently, 𝑀 is  semi-small compressible module if there exists a 

monomorphism : 𝑀 ⟶  𝑁 ,0 ≠ 𝑁 ≪𝑠𝑒𝑚 𝑀,  𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒 𝑀  is said to be semi-small 

retractable module if 𝐻𝑜𝑚(𝑀, 𝐾)  ≠ 0 , for every non-zero  semi-small sub module 𝐾in 𝑀. 

Equivalently, 𝑀 is semi-small retractable if there exists a homomorphism 𝑓: 𝑀 ⟶  𝑁 

whenever 0 ≠ 𝑁 ≪𝑠𝑒𝑚 𝑀 .  

    In this paper we introduce and study the concept of semi-small compressible 𝑚𝑜𝑑𝑢𝑙𝑒𝑠 and 

semi-small retractable 𝑚𝑜𝑑𝑢𝑙𝑒s as a generalization of compressible 𝑚𝑜𝑑𝑢𝑙𝑒 and retractable 

𝑚𝑜𝑑𝑢𝑙𝑒 respectively and give some of their advantages characterizations and examples.  

Keywords: compressible module, retractable module, small sub module, semi-small sub module, 

semi-small compressible module, semi-small retractable module. 

1. Introduction 

     Let R be a commutative ring with 1 and M be a left unitary   𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒. Authors that 

introduced and studied the concept of small sub modules where a proper sub module 𝑁 of an 

𝑅 −module 𝑀 is termed a small sub module (𝑁 ≪ 𝑀), if 𝑁 + 𝐿 ≠ 𝑀 for every sub module 𝐿 of 

doi.org/10.30526/36.4.3156 

Article history: Received 25 December 2022, Accepted 26 February 2023, Published in October 2023 

 

 

 

Ibn Al-Haitham Journal for Pure and Applied Sciences 

Journal homepage: jih.uobaghdad.edu.iq 

 

Semi-Small Compressible Modules and Semi-Small Retractable Modules 

 
 Mohammed Baqer Hashim Al Hakeem*  

Department of   Mathematics, College of Science, 

Baghdad University, Baghdad, Iraq. 

 

    Nuhad S. Al-Mothafar  

Department of   Mathematics, College of Science, 

Baghdad University, Baghdad, Iraq. 

 
Maryam Haghjooyan  

Ministry of Education, Tazkiyeh High School, 

Motahari St. Darya St., Saadat Abad, Tehran, Iran. 

 

https://creativecommons.org/licenses/by/4.0/
mailto:mbhmsc2015110@gmail.com
mailto:mbhmsc2015110@gmail.com
mailto:nuhad.salim@sc.uobaghdad.edu.iq
mailto:haghjooyanmaryam@gmail.com


IHJPAS. 36 (4) 2023 

408 
 

𝑀[1]. A proper sub module 𝑁 of 𝑀 is said to be primary if whenever 𝑟 ∈ 𝑅 , 𝑚 ∈ 𝑀 with 𝑟𝑚 ∈

𝑁 implies either 𝑚 ∈ 𝑁 or 𝑟𝑛 ∈ [𝑁: 𝑀] for some positive integer 𝑛 , where [𝑁: 𝑀] = {𝑟 ∈

𝑅: 𝑟𝑀 ⊆ 𝑁}[2] . In [3] Mijbas and K. Abdullah introduced and studied the concept of semi-small 

sub modules , where a sub module 𝑁 of an 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒 𝑀 is termed  semi-small sub module 

𝑁 ≪𝑠𝑒𝑚 𝑀 if 𝑁 + 𝐵 ≠ 𝑀 for any primary sub module 𝐵 of 𝑀. An 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒 𝑀 is termed 

compressible if 𝑀 can be embedded in every non-zero sub module in 𝑀,[4]. An  𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒 𝑀  

is said to be semi-small compressible if 𝑀 can be embedded in every non-zero semi-small sub 

module of𝑀. Equivalently, 𝑀 is semi-small compressible if there exists a monomorphism  

𝑓: 𝑀 ⟶  𝑁 whenever0 ≠ 𝑁 ≪𝑠𝑒𝑚 𝑀. 

    Under which condition we introduce and study the concept of semi-small compressible as a 

generalization of compressible module, and we give some properties, characterization and 

examples. In addition, we see that under condition semi-small compressible, small compressible 

and compressible are equivalent.  An 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒 𝑀 is said to be semi-small retractable module 

if (𝑀, 𝐾)  ≠ 0 , for every non-zero semi-small sub module 𝐾of 𝑀, some of their advantages 

characterizations and examples are given. We also study the relation between semi-small 

compressible, semi-small retractable module and some of classes of modules. 

 

𝟐. 𝐏𝐫𝐞𝐥𝐢𝐦𝐢𝐧𝐚𝐫𝐢𝐞𝐬 

𝐃𝐞𝐟𝐢𝐧𝐢𝐭𝐢𝐨𝐧 (𝟐. 𝟏): Let 𝑀 be an 𝑅 −module and 𝑁 ≤ 𝑀:  

(1) 𝑁 is termed small submodule of 𝑀 , (𝑁 ≪ 𝑀)  if 𝑁 + 𝐾 = 𝑀 implies 𝐾 = 𝑀 , for any sub module 

𝐾 of 𝑀[1].  

(2) An 𝑅 −module𝑀 is termed hollow if every proper sub module of 𝑀 is small[5]. 

(3) A proper submodule𝑁 of 𝑀 is termed primary if whenever 𝑟 ∈ 𝑅 , 𝑚 ∈ 𝑀 such that  𝑟 . 𝑚 ∈ 𝑁 

implies either 𝑚 ∈ 𝑁 or 𝑟𝑛 ∈ [𝑁: 𝑀] for some positive integer 𝑛 , where [𝑁: 𝑀] = {𝑟 ∈ 𝑅: 𝑟𝑀 ⊆

𝑁} [2] 

(4) A proper sub module 𝑁 is termed semi-small sub module of 𝑀 , (𝑁 ≪𝑠𝑒𝑚 𝑀)  if  𝑁 + 𝑃 ≠ 𝑀 , for 

any primary submodule 𝑃 of  𝑀, [3]. 

𝐑𝐞𝐦𝐚𝐫𝐤𝐬 𝐚𝐧𝐝 𝐞𝐱𝐚𝐦𝐩𝐥𝐞𝐬 (𝟐. 𝟐): [𝟑]  

(1)  (6̅)  is a semi-small sub𝑚𝑜𝑑𝑢𝑙𝑒  of 𝑍12  as 𝑍 − 𝑚𝑜𝑑𝑢𝑙𝑒. 

(2)  (0) is the only semi-small sub module If 𝑀 is a semi-simple 𝑚𝑜𝑑𝑢𝑙𝑒. 

(3)  (2̅) and (3̅) are not semi-small sub module of 𝑍6. 

(4) Each small sub module is semi-small. However, conversely is true or not in general. 

(5) Let 𝑁 be  a proper sub module in 𝑀. If W⊂ 𝑁 ≪𝑠𝑒𝑚 𝑀. Therefore W≪𝑠𝑒𝑚 𝑀 

(6) Let 𝑀, 𝑀′ be 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒𝑠 and 𝜓: 𝑀 ⟶ 𝑀′ be an 𝑅 − ℎ𝑜𝑚𝑜𝑚𝑜𝑟𝑝ℎ𝑖𝑠𝑚.          

If 𝐴 ≪𝑠𝑒𝑚 𝑀 with  𝑘𝑒𝑟 𝜓 ≤ 𝐴, then 𝜓(𝐴) ≪𝑠𝑒𝑚 𝑀′.  

3. Semi-Small Compressible Modules 

       In this section, we introduce the concept of semi-small compressible 𝑚𝑜𝑑𝑢𝑙𝑒 as a 

generalization of compressible 𝑚𝑜𝑑𝑢𝑙𝑒. Give some of it is basic properties, examples and 

characterizations of this concept.   

 



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409 
 

Definition (3.1): An  𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒 𝑀  is said to be semi-small compressible if 𝑀 can be 

embedded in every non-zero semi-small sub module of  𝑀. Equivalently, 𝑀 is semi-small 

compressible if there exists a monomorphism 𝑓: 𝑀 ⟶  𝑁 whenever                      0 ≠ 𝑁 ≪𝑠𝑒𝑚 𝑀.  

Remarks and Examples (3.2): 

1. It is obvious that every compressible module is semi-small compressible 𝑚𝑜𝑑𝑢𝑙𝑒, but the 

converse is not true. 

2.  𝑍6 as Z-module is not semi-small compressible since (0̅) is the only semi-small sub module, 

see, [3]. 

3. 𝑍 𝑎𝑠 𝑍 − 𝑚𝑜𝑑𝑢𝑙𝑒 is semi-small compressible module, because it is compressible 𝑚𝑜𝑑𝑢𝑙𝑒, 

see[4]. 

4. If 𝑎𝑛 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒𝑀 is semi-simple, then 𝑀 is not semi-small compressible  module               

(Because (0) is the only semi-small sub module in 𝑀).  

5. Every simple 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒  is semi-small compressible module but not conversely, because 

𝑍 𝑎𝑠 𝑍 − 𝑚𝑜𝑑𝑢𝑙𝑒 is a semi-small compressible 𝑚𝑜𝑑𝑢𝑙𝑒 but not simple.  

6. 𝑍12 𝑎𝑠 𝑍 − 𝑚𝑜𝑑𝑢𝑙𝑒 is not semi-small compressible. (Because 𝑍12 cannot be embedded in 〈6̅〉 

and  〈6̅〉 ≪𝑠𝑒𝑚 𝑍12 ). In addition 𝑄 𝑎𝑠 𝑍 − 𝑚𝑜𝑑𝑢𝑙𝑒 is not semi-small compressible module, 

since 𝐻𝑜𝑚𝑅(𝑄, 𝑍) = 0 , where 𝑍 ≪𝑠𝑒𝑚 𝑄. (Since every finitely generated sub module of 𝑄 is 

semi-small sub module in 𝑄.   

7. A homomorphic image of a semi-small compressible 𝑚𝑜𝑑𝑢𝑙𝑒 need not be                                        

semi-small compressible in general for example 𝑍 𝑎𝑠 𝑍 − 𝑚𝑜𝑑𝑢𝑙𝑒 is a semi-small 

compressible module and 
𝑧

12𝑧
≃ 𝑧12 is not semi-small compressible module see (5). 

Proposition (3.3): A semi-small sub module of semi-small compressible 𝑚𝑜𝑑𝑢𝑙𝑒 is also semi-

small compressible 𝑚𝑜𝑑𝑢𝑙𝑒. 

Proof: Let 0 ≠ 𝐾 ≪𝑠𝑒𝑚 𝑀 and 𝑀 be semi-small compressible 𝑚𝑜𝑑𝑢𝑙𝑒 and let                0 ≠ 𝐿 ≤

𝐾 ≪𝑠𝑒𝑚 
𝑀 , then  𝐿 ≪𝑠𝑒𝑚 𝑀 [3]. Since 𝑀 is semi-small compressible, so ∃  a monomorphism 

𝑓: 𝑀 ⟶ 𝐿 and 𝑖: 𝐾 ⟶ 𝑀 is the inclusion homomorphism, then                    𝑓 ∘ 𝑖: 𝐾 ⟶ 𝐿 is a 

monomorphism. Therefore 𝐾 is a semi-small compressible 𝑚𝑜𝑑𝑢𝑙𝑒. 

Proposition (3.4): Let 𝑀1 and 𝑀2 be isomorphic 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒𝑠. Then 𝑀1is semi-small 

compressible if and only if 𝑀2is semi-small compressible.   

Proof: Suppose that 𝑀2 is semi-small compressible and let 𝜙: 𝑀1 ⟶ 𝑀2 be an isomorphism. Let 0 ≠

𝑁 ≪𝑠𝑒𝑚 𝑀1, then by[3] 0 ≠ 𝜙(𝑁) ≪𝑠𝑒𝑚 𝑀2. Put                                       𝐾 = 𝜙(𝑁) ≪𝑠𝑒𝑚 𝑀2, so 

𝛼: 𝑀2 ⟶ 𝐾 is a monomorphism (by assumption), let ℎ =   𝜙−1 │𝐾 , then 𝑔: 𝐾 ⟶ 𝑀1 is a 

monomorphism. 𝑔(𝐾) =  𝜙−1(𝜙(𝑁)) = 𝑁. Hence, we have a composition. Let 𝜓 = ℎ ∘  𝛼 ∘ 𝜙.  

Hence, 𝜓: 𝑀1 ⟶ 𝑁 is a monomorphism. Therefore 𝑀1is semi-small compressible 𝑚𝑜𝑑𝑢𝑙𝑒.  

Remark (3.5):  The direct sum of semi-small compressible 𝑚𝑜𝑑𝑢𝑙𝑒 need not be semi-small 

compressible. Consider the following example, let𝑍6 = 𝑍3⨁𝑍2  as 𝑍 − 𝑚𝑜𝑑𝑢𝑙𝑒. 𝑍3, 𝑍3 are semi-

small compressible modules, but 𝑍6 is not semi-small compressible module see remarks and 

examples (3.2) point (2).   

       An  𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒 𝑀  is said to be small compressible if 𝑀 can be embedded in every nonzero 

small sub module of 𝑀. Equivalently, 𝑀 is small compressible if there exists a monomorphism 

𝑓: 𝑀 ⟶  𝑁 whenever  0 ≠ 𝑁 ≪ 𝑀[4]. 

 

𝐑𝐞𝐦𝐚𝐫𝐤 (𝟑. 𝟔):  



IHJPAS. 36 (4) 2023 

410 
 

Every semi-small compressible module is small compressible module. 

𝑷𝒓𝒐𝒐𝒇:  Let 0 ≠ 𝑁 ≪ 𝑀, then by [3]  𝑁 ≪𝑠𝑒𝑚 𝑀 and 𝑀 is semi-small compressible 𝑚𝑜𝑑𝑢𝑙𝑒, 

therefor 𝑀 is small compressible module. Conversely is not true for instance 𝑍6 as Z-module is small 

compressible, [6]. However, not semi-small compressible see remarks and examples (3.2) point (2). 

   𝑷𝒓𝒐𝒑𝒐𝒔𝒊𝒕𝒊𝒐𝒏 (𝟑. 𝟕): Let 𝑀 be a finitely generated (or multiplication)  𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒 . Then 𝑀 

is small compressible if and only if 𝑀is semi-small compressible. 

𝑷𝒓𝒐𝒐𝒇: Let 𝑁 ≪𝑠𝑒𝑚 𝑀. We want to show that 𝑀is semi-small compressible. Since 𝑀 is finitely 

generated ((or multiplication), then by proposition (1.3)[3], so 𝑁 ≪ 𝑀, but 𝑀 is small 

compressible 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒. Therefore 𝑀 is semi-small compressible. Conversely clear by remark 

(3.6). 

   𝑷𝒓𝒐𝒑𝒐𝒔𝒊𝒕𝒊𝒐𝒏 (𝟑. 𝟖): Let 𝑀 be a hollow 𝑚𝑜𝑑𝑢𝑙𝑒. Then the following statements are 

equivalent: 

(1) 𝑀 is compressible 𝑚𝑜𝑑𝑢𝑙𝑒. 

(2) 𝑀 is semi-small compressible 𝑚𝑜𝑑𝑢𝑙𝑒. 

(3) small compressible 𝑚𝑜𝑑𝑢𝑙𝑒 . 

Proof:  (1) ⟹ (2) It is clear by remarks and examples (3.2) point (1).  

              (2) ⟹ (3) It is clear by remark (3.6). 

(3) ⟹ (1)Let𝐾 ≤ 𝑀. Since 𝑀 is ℎ𝑜𝑙𝑙𝑜𝑤  𝑚𝑜𝑑𝑢𝑙𝑒 and small compressible 𝑚𝑜𝑑𝑢𝑙𝑒, then ∃ a 

monomorphism  𝑓: 𝑀 ⟶ 𝐾. Therefor  𝑀 is compressible 𝑚𝑜𝑑𝑢𝑙𝑒. 

      An 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒 𝑀 is termed quasi-Dedekind 𝑚𝑜𝑑𝑢𝑙𝑒 if for all  𝑓 ∈ 𝐸𝑛𝑑𝑅(𝑀) ,                 𝑓 ≠ 0 

implies 𝐾𝑒𝑟𝑓 = 0. [7] 

      An 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒 𝑀 is termed small quasi-Dedekind 𝑚𝑜𝑑𝑢𝑙𝑒 if for all  𝑓 ∈ 𝐸𝑛𝑑𝑅(𝑀), 𝑓 ≠ 0 

implies 𝐾𝑒𝑟𝑓 ≪ 𝑀. [7]. 

   We introduce the following  

     Definition (3.9): An 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒 𝑀 is termed semi-small quasi-Dedekind 𝑚𝑜𝑑𝑢𝑙𝑒 if for all  𝑓 ∈

𝐸𝑛𝑑𝑅(𝑀) , 𝑓 ≠ 0 implies 𝐾𝑒𝑟𝑓 ≪𝑠𝑒𝑚 𝑀. 

 Remarks(3.10):  

1. It is clear that every quasi-Dedekind𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒 is semi-small quasi-Dedekind, but not 

conversely. 

2. Every small quasi-Dedekind is semi-small quasi-Dedekind, but not conversely. 

Proof: Let 0 ≠ 𝑓 ∈ 𝐸𝑛𝑑𝑅(𝑀), where 𝑀 be an 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒 since 𝑀 is a small quasi-Dedekind, 

then 𝐾𝑒𝑟𝑓 ≪ 𝑀, hence 𝐾𝑒𝑟𝑓 ≪𝑠𝑒𝑚 𝑀. Thus 𝑀 is a semi-small quasi-Dedekind 𝑚𝑜𝑑𝑢𝑙𝑒. 

We introduce an 𝑆 − ℎ𝑜𝑙𝑙𝑜𝑤 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒 as a generalization of a hollow module. 

Definition (3.11): An 𝑅 −module𝑀 is termed S-hollow if every proper submodule in 𝑀 is semi-

small. 

Remarks (3.12):  

(1) Every hollow module is S-hollow module, but the converse is not true. 

(2) Every simple module is S-hollow module, but not conversely for example 𝑍4 as 

𝑍 −module is S-hollow module, but not simple. 

(3) If an 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒𝑀 is a semi-simple, then 𝑀 is not S-hollow module (since (0) is the 

only semi-small sub module). 



IHJPAS. 36 (4) 2023 

411 
 

(4)  𝑍12 as 𝑍 −module is not S-hollow module, because (3̅) is not semi-small sub module in 

𝑍12 ,[3]. 

Proposition (3.13): Let 𝑀 be an S-hollow 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒. Then 𝑀 is compressible module if and 

only if  𝑀 is a semi-small compressible module. 

Proof: let 0 ≠ 𝑁 ≤ 𝑀 and  𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒 𝑀 be S-hollow, then 𝑁 ≤𝑠𝑒𝑚 𝑀, but 𝑀 is a semi-small 

compressible module, thus there exists a monomorphism 𝑓: 𝑀 ⟶  𝑁. Therefore 𝑀 is 

compressible module. Conversely clear by remarks and examples (3.2) point (1).  

  

4. Semi-Small Retractable Modules 

       We introduce the concept of semi-small retractable 𝑚𝑜𝑑𝑢𝑙𝑒 as a generalization of retractable 

𝑚𝑜𝑑𝑢𝑙𝑒. Give some of basic properties, examples and characterizations of this concept.   

Definition (4.1): An  𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒 𝑀  is said to be semi-small retractable if (𝑀. 𝐾)  ≠ 0 , for 

every non-zero semi-small sub module 𝐾of 𝑀. Equivalently, 𝑀 is semi-small retractable if there 

exists a homomorphism 𝑓: 𝑀 ⟶  𝑁 whenever 0 ≠ 𝑁 ≪𝑠𝑒𝑚 𝑀. 

Remarks and Examples (4.2): 

1. It is obvious that every semi-small compressible module is semi-small retractable 𝑚𝑜𝑑𝑢𝑙𝑒. 

However, conversely is not true for instance 𝑧12 is semi-small retractable but not semi-small 

compressible 𝑚𝑜𝑑𝑢𝑙𝑒 see remarks and examples (3.2) point (7). 

2. 𝑍 𝑎𝑠 𝑍 − 𝑚𝑜𝑑𝑢𝑙𝑒 is semi-small retractable 𝑚𝑜𝑑𝑢𝑙𝑒, because it is semi-small 

compressible 𝑚𝑜𝑑𝑢𝑙𝑒. 

3. Every simple 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒  is semi-small retractable module but not conversely, because 

𝑍 𝑎𝑠 𝑍 − 𝑚𝑜𝑑𝑢𝑙𝑒 is a semi-small retractable 𝑚𝑜𝑑𝑢𝑙𝑒 but not simple.  

4. Every retractable 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒 is semi-small retractable 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒, but the converse is 

not true ingeneral. 

5.  Every semi-simple 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒 is semi-small retractable because it is retractable. 

6. Every compressible 𝑚𝑜𝑑𝑢𝑙𝑒 is semi-small retractable 𝑚𝑜𝑑𝑢𝑙𝑒, but the converse is not 

true for instance 𝑧6 is semi-small retractable but not compressible 𝑚𝑜𝑑𝑢𝑙𝑒 see,[4]. 

Proposition (4.3): A semi-small sub module of semi-small retractable 𝑚𝑜𝑑𝑢𝑙𝑒 is also semi-small 

retractable 𝑚𝑜𝑑𝑢𝑙𝑒. 

Proof: Let 0 ≠ 𝐾 ≪𝑠𝑒𝑚 𝑀 and 𝑀 be semi-small retractable module. Let  0 ≠ 𝐿 ≤ 𝐾 ≪𝑠𝑒𝑚 
𝑀 , 

by [3]   𝐿 ≪𝑠𝑒𝑚 𝑀 . Since 𝑀 is semi-small retractable, so ∃  a homomorphism 𝑓: 𝑀 ⟶ 𝐿 and 

𝑖: 𝐾 ⟶ 𝑀 is the inclusion homomorphism, then 𝑓 ∘ 𝑖: 𝐾 ⟶ 𝐿 be a homomorphism. Therefore 𝐾 

is a semi-small retractable 𝑚𝑜𝑑𝑢𝑙𝑒. 

Proposition (4.4): Let 𝑀1 and 𝑀2 be isomorphic 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒𝑠. Then 𝑀1is semi-small 

retractable if and only if 𝑀2is semi-small retractable   

Proof: Suppose that 𝑀2 is semi-small retractable and let 𝑓: 𝑀1 ⟶ 𝑀2 be an isomorphism. Let 0 ≠

𝑁 ≪𝑠𝑒𝑚 𝑀1, then by[3] 0 ≠ 𝑓(𝑁) ≪𝑠𝑒𝑚 𝑀2. Put 𝐾 = 𝑓(𝑁) ≪𝑠𝑒𝑚 𝑀2, so  𝜃: 𝑀2 ⟶ 𝐾 is a 

homomorphism (by assumption), let  𝑔 =   𝑓−1 │𝐾 , then 𝑔: 𝐾 ⟶ 𝑀1 is a homomorphism. 𝑔(𝐾) =

 𝑓−1(𝑓(𝑁)) = 𝑁, hence, we have a composition.                                 Let  Η = 𝑔 ∘ 𝜃 ∘ 𝑓. Hence 

Η: 𝑀1 ⟶ 𝑁 is a homomorphism. Therefore 𝑀1is semi-small retractable 𝑚𝑜𝑑𝑢𝑙𝑒.  

Proposition (4.5): Let 𝑀 be an  𝑆 − ℎ𝑜𝑙𝑙𝑜𝑤 𝑚𝑜𝑑𝑢𝑙𝑒, then the following are equivalent  

(1)  𝑀 is retractable 𝑚𝑜𝑑𝑢𝑙𝑒. 

(2) 𝑀 is semi-small retractable 𝑚𝑜𝑑𝑢𝑙𝑒. 



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Proof: (1) ⟹ (2) clearly by remarks and examples (4.2) point (4).  

(2) ⟹ (1)Let 0 ≠ 𝑁 ≤ 𝑀 and  𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒 𝑀 be S-hollow, then 𝑁 ≤𝑠𝑒𝑚 𝑀, but 𝑀 is a semi-

small retractable module. Thus, 𝐻𝑜𝑚(𝑀, 𝑁)  ≠ 0 .Therefore 𝑀 is retractable module.  

Proposition (4.6): If 𝑀 is semi-small quasi-Dedekind 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒, then 𝑀 cannot be semi-small 

retractable. 

Proof: Suppose that 𝑀 is semi-small quasi-Dedekind 𝑚𝑜𝑑𝑢𝑙𝑒 and let𝑁 = 𝐾𝑒𝑟𝑓 ≤ 𝑀, but 𝑀 is 

semi-small quasi-Dedekind, then𝐾𝑒𝑟𝑓 ≪𝑠𝑒𝑚 𝑀,𝑓 ≠ 0, thus 𝐻𝑜𝑚(𝑀, 𝐾𝑒𝑟𝑓) = 0.  Therefore 𝑀 

cannot be semi-small retractable 𝑚𝑜𝑑𝑢𝑙𝑒.  

    Recall that an 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒 𝑀 is termed monoform if for each non-zero sub module 𝑁 of 𝑀 and    

  for each 𝑓 ∈ 𝐻𝑜𝑚𝑅(𝑁, 𝑀), 𝑓 ≠ 0  implies 𝐾𝑒𝑟𝑓 = 0, [8]. 

  Definition (4.7): An 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒 𝑀 is termed semi-small monoform if for each non-zero sub  

  module 𝑁 of 𝑀 and for each 𝑓 ∈ 𝐻𝑜𝑚𝑅(𝑁, 𝑀), 𝑓 ≠ 0  implies 𝐾𝑒𝑟𝑓 ≪𝑠𝑒𝑚  𝑁. 

  Remarks and examples (4.8):  

(1) Every semi-small compressible 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒 is semi-small monoform, but not 

conversely. For example, 𝑍4 as 𝑍 − 𝑚𝑜𝑑𝑢𝑙𝑒 is semi-small monoform but not semi-

small compressible.  

(2) Every semi-small quasi-Dedekind 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒 is semi-small monoform. However, 

not conversely. For example, 𝑍4 as 𝑍 − 𝑚𝑜𝑑𝑢𝑙𝑒 is semi-small monoform but not semi-

small quasi-Dedekind. 

  Proposition (4.9): Let 𝑀 be a quasi-Dedekind 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒. Then 𝑀 is semi-small monoform if  

   and only if 𝑀 is semi-small compressible.  

Proof: Suppose that 𝑀 is semi-small monoform. Let 0 ≠ 𝑁 ≪𝑠𝑒𝑚  𝑀, then 0 ≠ 𝑓 ∈

𝐻𝑜𝑚𝑅(𝑁, 𝑀). Since 𝑀 is quasi-Dedekind , then 𝑓 ∘ 𝑔: 𝑀 ⟶ 𝑁 ⟶ 𝑀 is a monomorphism, hence 

𝑔: 𝑀 ⟶ 𝑁 is a monomorphism. Thus 𝑀 is semi-small compressible. Conversely see remark (4.8) 

point (1). 

 

5. Conclusion 

In this work, the class of compressible and retractable modules have been generalized to new 

concepts called semi-small compressible and semi-small retractable modules. Several 

characteristics of this type of modules have been studied. Sufficient conditions under which these 

modules with compressible and retractable are discussed. 

In addition, we see relations between semi-small compressible modules and other related modules 

as semi-small retractable module semi-small quasi-Dedekind, semi-small monoform. 

 

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