IHJPAS. 36 (3) 2023 

365 

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 papers we 

introduced and studied concept P-small compressible 𝑚𝑜𝑑𝑢𝑙𝑒 (An  𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒 𝑀  is said to be 

P-small compressible if 𝑀 can be embedded in every of it is nonzero P-small submodule of 𝑀. 

Equivalently, 𝑀 is P-small compressible if there exists a monomorphism : 𝑀 ⟶  𝑁 ,0 ≠

𝑁 ≪𝑃 𝑀,  𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒 𝑀  is said to be P-small retractable if 𝐻𝑜𝑚(𝑀, 𝐾)  ≠ 0 , for every non-

zero P-small submodule 𝐾of 𝑀. Equivalently, 𝑀 is P-small retractable if there exists a 

homomorphism 𝑓: 𝑀 ⟶  𝑁 whenever 0 ≠ 𝑁 ≪𝑃 𝑀 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 submodule, P-small submodule, P-

small Compressible module, P-small Retractable module. Hollow module, PS-Hollow module. 

 

1. Introduction  

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

and studied concept small submodules. A proper submodule 𝑁 of an 𝑅 −module 𝑀 is termed a 

small submodule (𝑁 ≪ 𝑀), if 𝑁 + 𝐿 ≠ 𝑀 for every submodule 𝐿 of 𝑀[1]. A proper submodule 

𝑁 of 𝑀 is said to be prime if whenever    𝑟 ∈ 𝑅 , 𝑚 ∈ 𝑀 such that 𝑟. 𝑚 ∈ 𝑁 𝑖𝑚𝑝𝑙𝑖𝑒𝑠 either 𝑚 ∈ 𝑁 

or ∈ [𝑁: 𝑀] ; [𝑁: 𝑀] = {𝑟 ∈ 𝑅: 𝑟𝑀 ⊆ 𝑁}[2] . In [3] Iman M.A.Hadi and Tammader A.Ibrahiem 

introduced and studied the concept of P-small submodules , where a submodule 𝑁 of an 𝑅 −

𝑚𝑜𝑑𝑢𝑙𝑒 𝑀 is called  P-small submodule 𝑁 ≪𝑃 𝑀 if 𝑁 + 𝑃 ≠ 𝑀 for any prime submodule 𝑃 of 

𝑀. An 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒𝑀 is called compressible if 𝑀 can be embedded in every non-zero submodule. 

An  𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒 𝑀  is said to be P-small compressible if 𝑀 can be embedded in every of it is 

nonzero P-small submodule of 𝑀. Equivalently, 𝑀 is P-small compressible if there exists a 

monomorphism 𝑓: 𝑀 ⟶  𝑁 whenever 0 ≠ 𝑁 ≪𝑃 𝑀. 

doi.org/10.30526/36.3.3089 

Article history: Received 27 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 

 

P-small Compressible Modules and P-small Retractable Modules 

 

     Mohammed Baqer Hashim Al Hakeem*  

Department of  Mathematics, College of Science, 

Baghdad University, Baghdad, Iraq. 

 

mbhmsc2015110@gmail.com 

 

 

 

      Nuhad S. Al-Mothafar   

Department of  Mathematics, College of Science, 

Baghdad University, Baghdad, Iraq. 

 

nuhad.salim@sc.uobaghdad.edu.iq 
 

 

 

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


IHJPAS. 36 (3) 2023 

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     In this paper we introduce and study the concept of P-small compressible as a generalization of 

compressible module, and we give some properties , characterization and examples. In addition, 

we see that under condition. P-small compressible, small compressible and compressible are 

equivalent.some of their advantages characterizations and examples are given. We also study the 

relation between P-small compressible module, P-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 called small submodule of 𝑀 , (𝑁 ≪ 𝑀)  if 𝑁 + 𝐾 = 𝑀 implies 𝐾 = 𝑀 , for any 

submodule 𝐾 of 𝑀[1].  

2. An 𝑅 −module𝑀 is called hollow if every proper submodule is small in𝑀[4]. 

3. A proper submodule 𝑁 of 𝑀 is called prime if whenever 𝑟 ∈ 𝑅 , 𝑚 ∈ 𝑀 implies either 𝑚 ∈ 𝑁 

or 𝑟 ∈ [𝑁: 𝑀] : [𝑁: 𝑀] = {𝑟 ∈ 𝑅: 𝑟𝑀 ⊆ 𝑁} [2] 

4. A proper submodule 𝑁 is called P-small submodule of 𝑀 , (𝑁 ≪𝑃 𝑀)  if  𝑁 + 𝑃 ≠ 𝑀 , for any 

prime submodule 𝑃 of  𝑀, [3]. 

5. An 𝑅 −module𝑀 is called PS-hollow if every proper submodule in 𝑀 is P-small[3]. 

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

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

𝑓: 𝑀 ⟶  𝑁 whenever  0 ≠ 𝑁 ≪ 𝑀[5]. 

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

𝐾𝑒𝑟𝑓 = 0, [7]. 

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

implies 𝐾𝑒𝑟𝑓 ≪ 𝑀, [7]. 

Remark(2.2) :[3](1)  (2̅)   is P-small sub𝑚𝑜𝑑𝑢𝑙𝑒  of 𝑍6  as 𝑍 − 𝑚𝑜𝑑𝑢𝑙𝑒. 

                         (2) (2̅) and (3̅) are not P-small submodule of 𝑍6. 

(3) If 𝑀 is semi-simple 𝑚𝑜𝑑𝑢𝑙𝑒, then (0) is the only P-small submodule. 

 

Remark (2.3): Each small submodule is P-small. But the converse is not true in general for 

example(2̅) is P-small sub𝑚𝑜𝑑𝑢𝑙𝑒  of 𝑍6  as 𝑍 − 𝑚𝑜𝑑𝑢𝑙𝑒 but not small. 

 

Lemma (2.4):  

1. Let 𝑁 be a proper submodule of 𝑀. If W⊂ 𝑁 ≪𝑃 𝑀, then W≪𝑃 𝑀. In particular if  W is a 

direct summand of 𝑁 and 𝑁 ≪𝑃 𝑀, then W≪𝑃 𝑀. 

2. Let 𝑁1 and 𝑁2 be proper submodules of 𝑀. If 𝑁1 + 𝑁2 ≪𝑃 𝑀, then 𝑁1 ≪𝑃 𝑀, 𝑁2 ≪𝑃 𝑀, the 

converse is not true. 

3. Let 𝐴 ⊂ 𝐵 ⊂ 𝑁 ⊂ 𝑀. If B≪𝑃 𝑁, then 𝐴 ≪𝑃 𝑀. 

4. Let 𝑀, 𝑀′ be 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒𝑠 and 𝜓: 𝑀 ⟶ 𝑀′ be an 𝑅 − ℎ𝑜𝑚𝑜𝑚𝑜𝑟𝑝ℎ𝑖𝑠𝑚. If 𝐴 ≪𝑃 𝑀, then 

𝜓(𝐴) ≪𝑃 𝑀′. 

 

3 .P-small Compressible Module 

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

of compressible 𝑚𝑜𝑑𝑢𝑙𝑒, give some of basic properties, examples and characterizations of this 

concept.   



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

every of it is nonzero P-small submodule of 𝑀. Equivalently, 𝑀 is P-small compressible if there 

exists a monomorphism 𝑓: 𝑀 ⟶  𝑁 whenever                           0 ≠ 𝑁 ≪𝑃 𝑀.  

Remarks and examples (3.2): 

1. It’s obvious that every compressible module is P-small compressible 𝑚𝑜𝑑𝑢𝑙𝑒, but the converse 

is not true. For example 𝑍6 as Z-module is P-small compressible since (0̅) is the only P-small 

submodule, but not compressible. 

2. 𝑍 𝑎𝑠 𝑍 − 𝑚𝑜𝑑𝑢𝑙𝑒 is  P-small compressible module, because it's compressible 𝑚𝑜𝑑𝑢𝑙𝑒. 

3. 𝑍𝑃  𝑎𝑠 𝑍 − 𝑚𝑜𝑑𝑢𝑙𝑒 is P-small compressible 𝑚𝑜𝑑𝑢𝑙𝑒; 𝑃 𝑖𝑠 𝑎 𝑝𝑟𝑖𝑚𝑒 𝑛𝑢𝑚𝑏𝑒𝑟. 

4. Every simple 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒  is P-small compressible module but not conversely, because 

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

5. 𝑍4 𝑎𝑠 𝑍 − 𝑚𝑜𝑑𝑢𝑙𝑒 is not P-small compressible.(Because 𝑍4 can’t be embedded in 〈2̅〉 

and  〈2̅〉 ≪𝑃 𝑍4 ) . 

6. A homomorphic image of a P-small compressible 𝑚𝑜𝑑𝑢𝑙𝑒 need not be                                        P-

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

and 
𝑧

4𝑧
≃ 𝑧4 is not P-small compressible module view remark (5). 

Proposition(3.3): A P-small submodule of  P-small compressible 𝑚𝑜𝑑𝑢𝑙𝑒 is also P-small 

compressible 𝑚𝑜𝑑𝑢𝑙𝑒. 

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

then  𝐿 ≪𝑃 𝑀 [3]. Since 𝑀 is P-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 P-small compressible 𝑚𝑜𝑑𝑢𝑙𝑒. 

Proposition(3.5): If an 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒 𝑀 has no prime submodule such that ∃ a monomorphism 

𝑓: 𝑀 ⟶ 𝑁 , ∀𝑁 ⊊ 𝑀 , then 𝑀 is  P-small compressible .  

Proof: Suppose 𝑀 has no prime submodule and let 𝑁 ⊊ 𝑀, then 𝑁 ≪𝑃 𝑀 ⌈3⌉and by assumption 

 𝑀 is  P-small compressible.  

Proposition(3.6): Let 𝑀1 and 𝑀2 be isomorphic 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒𝑠. Then 𝑀1is P-small compressible 

if and only if 𝑀2is P-small compressible. 

 Proof: Let 0 ≠ 𝑁 ≪𝑃 𝑀1 and suppose that 𝑀2 is P-small compressible. Let 𝜙: 𝑀1 ⟶ 𝑀2 be an 

isomorphism., then by[3] 0 ≠ 𝜙(𝑁) ≪𝑃 𝑀2.Put 𝐾 = 𝜙(𝑁) ≪𝑃 𝑀2, so 𝛼: 𝑀2 ⟶ 𝐾 is a 

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

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

𝜓: 𝑀1 ⟶ 𝑁 is a monomorphism. Therefore 𝑀1is P-small compressible 𝑚𝑜𝑑𝑢𝑙𝑒.     

Remark(3.7):  The direct sum of P-small compressible 𝑚𝑜𝑑𝑢𝑙𝑒 need not be P-small compressible. 

Consider the following example let 𝑍4 ≃ 𝑍2⨁𝑍2  as Z-module . 𝑍2 is  P-small compressible 

module, but 𝑍4 is not P-small compressible module see remarks and examples (2.3) point(5) 

  Proposition(3.8):Let 𝑀 = 𝑀1⨁𝑀2 be an 𝑅 −module such that 𝑎𝑛𝑛𝑅 𝑀1⨁𝑎𝑛𝑛𝑅 𝑀2 = 𝑅. 

If 𝑀1 and  𝑀2 are P-small compressible 𝑚𝑜𝑑𝑢𝑙𝑒𝑠, then 𝑀 is P-small compressible. 

 Proof:  Let 0 ≠ 𝑁 = 𝐾1⨁𝐾2 ≪𝑃 𝑀. Then by theorem (1.12)[3] .0 ≠ 𝐾1 ≪𝑃 𝑀1 ≤ 𝑀  and 0 ≠

𝐾2 ≪𝑃 𝑀2 ≤ 𝑀. But 𝑀1and 𝑀2 P-small compressible 𝑚𝑜𝑑𝑢𝑙𝑒𝑠, so ∃ monomorphisms 𝑓: 𝑀1 ⟶

𝐾1and 𝑔: 𝑀2 ⟶ 𝐾2 . Define 𝜓: 𝑀 ⟶ 𝑁 by  𝜓(𝑎, 𝑏) = (𝑓(𝑎), 𝑔(𝑏)), it can be easily show that 

𝜓is a monomorphism.  Therefore 𝑀 is P-small compressible.   



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Proposition(3.9):Let 𝑀 = 𝑀1⨁𝑀2 be P-small compressible 𝑚𝑜𝑑𝑢𝑙𝑒 such that 

𝑎𝑛𝑛𝑅 𝑀1⨁𝑎𝑛𝑛𝑅 𝑀2 = . 0 ≠ 𝐾1 ≪𝑃 𝑀1 ≤ 𝑀 and 0 ≠ 𝐾2 ≪𝑃 𝑀2 ≤ 𝑀 with             𝑁 =

𝐾1⨁𝐾2 ≪𝑃 𝑀, then 𝑀1 and 𝑀2 are P-small compressible 𝑚𝑜𝑑𝑢𝑙𝑒𝑠 .   

Proof:  Let 0 ≠ 𝐾1 ≤  𝑁 = 𝐾1⨁𝐾2 ≪𝑃 𝑀, then by remarks and examples(1.2)(7)[3] 𝐾1 ≪𝑃 𝑀, 

but 𝑀 be P-small compressible 𝑚𝑜𝑑𝑢𝑙𝑒, so ∃ a monomorphisms                       𝑓: 𝑀 ⟶ 𝐾1 and 

𝐽: 𝑀1 ⟶ 𝑀1⨁𝑀2 = 𝑀, hence we have a composition . Let𝜓 = 𝑓 ∘ 𝐽, thus 𝜓 : 𝑀1 ⟶ 𝐾1 is a 

monomorphism . Therefore 𝑀1is P-small compressible 𝑚𝑜𝑑𝑢𝑙𝑒. 

The same way we can prove 𝑀2 is P-small compressible 𝑚𝑜𝑑𝑢𝑙𝑒 .  

        

Remarks and  Examples (3.10):  

1.  Every P-small compressible module is small compressible module. 

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

therefor 𝑀 is small compressible module. 

2. 𝑍6as 𝑍 − 𝑚𝑜𝑑𝑢𝑙𝑒 is small compressible, since (0̅) the only P-small submodule of 𝑍6. 

3. 𝑄 𝑎𝑠 𝑍 − 𝑚𝑜𝑑𝑢𝑙𝑒 is not P-small compressible module, since 𝐻𝑜𝑚𝑅 (𝑄, 𝑍) = 0 , where 𝑍 ≪𝑃 𝑄. 

Proposition(3.11): Let  𝑀 be an 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒 and 0 ≠ 𝑚 ∈ 𝑀 such that 𝑅𝑚 ⊊ 𝑀, then 𝑀 is 

small compressible if and only if  𝑀is P-small compressible. 

Proof: Suppose that 𝑀 is small compressible 𝑚𝑜𝑑𝑢𝑙𝑒 and let 𝑁 ≪𝑃 𝑀 , then by [3]  

𝑁 ≪ 𝑀 and since 𝑀 is small compressible 𝑚𝑜𝑑𝑢𝑙𝑒, therefore 𝑀 is P-small compressible. 

Conversely it's clear by remarks and examples (3.10)point(1)  

Corollary(3.12): A small compressible 𝑚𝑜𝑑𝑢𝑙𝑒 𝑀 is P-small compressible, if every cyclic 

submodule of 𝑀 is P-small submodule in 𝑀 . 

Proof: obviously by above proposition. 

Proposition(3.13): Let 𝑀 be a finitely generated (or multiplication) 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒 . Then 𝑀 is 

small compressible if and only if 𝑀is P-small compressible. 

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

generated (or multiplication), then by proposition(1.4)[3], so 𝑁 ≪ 𝑀, but 𝑀 is small compressible 

𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒. Therefore 𝑀 is P-small compressible. Conversely clear by remarks and examples 

(3.10) point (1). 

Corollary(3.14): Let 𝑀 be a noetherian 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒. Then 𝑀 is small compressible if and only 

if 𝑀is P-small compressible. 

Proof: Since 𝑀 is noetherian, then every submodule is finitely  generated, then the result follows 

by proposition(3.13). Therefore 𝑀 is small compressible. Conversely  clear by remarks and 

examples (3.10) point (1). 

      Recall that an 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒 𝑀 is called almost finitely generated if 𝑀 is not finitely generated 

and every proper submodule of of 𝑀 is finitely generated[6]. 

Proposition(3.15): Let 𝑀 be an almost finitely generated 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒. Then 𝑀is P-small 

compressible if and only if 𝑀is small compressible. 

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

generated[6], then by corollary (1.11)[3], we get 𝑁 ≪ 𝑀, but 𝑀 is small compressible 𝑅 −

𝑚𝑜𝑑𝑢𝑙𝑒. Therefore 𝑀 is P-small compressible. Conversely  clear by remarks and examples (3.10) 

point (1). 

Proposition(3.16): Let 𝑀 be a hollow 𝑚𝑜𝑑𝑢𝑙𝑒. Then the following  statements are equivalent: 

(1) 𝑀 is compressible 𝑚𝑜𝑑𝑢𝑙𝑒. 



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(2) 𝑀 is P-small compressible 𝑚𝑜𝑑𝑢𝑙𝑒. 

(3) small compressible 𝑚𝑜𝑑𝑢𝑙𝑒 . 

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

              (2) ⟹ (3) It's clear by remarks (3.10) point (1) 

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

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

 

       

We introduce the following  

Definition (3.17): An 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒 𝑀 is called P-small quasi-Dedekind 𝑚𝑜𝑑𝑢𝑙𝑒 if for all ∈ (𝑀) 

, 𝑓 ≠ 0 implies 𝐾𝑒𝑟𝑓 ≪𝑃 𝑀. 

Remark (3.18): It's clear that every quasi-Dedekind is P-small quasi-Dedekind. 

Proposition(3.19): If 𝑀 is P-small quasi-Dedekind 𝑚𝑜𝑑𝑢𝑙𝑒, then 𝑀 can't be compressible . 

Proof: Suppose that 𝑀 is  P-small quasi-Dedekind 𝑚𝑜𝑑𝑢𝑙𝑒 and let 𝑁 = 𝐾𝑒𝑟𝑓 ≤ 𝑀, but 𝑀 is P-

small quasi-Dedekind, then 𝐾𝑒𝑟𝑓 ≪𝑃 𝑀,  𝑓 ≠ 0 , thus can't be embedded 𝑀 in  𝐾𝑒𝑟𝑓 , because 

𝐻𝑜𝑚(𝑀, 𝐾𝑒𝑟𝑓) = 0. Therefore 𝑀 can't be compressible 𝑚𝑜𝑑𝑢𝑙𝑒. 

Remark(3.20): 

Every small quasi-Dedekind is P-small quasi-Dedekind. 

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

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

 

4. P-small Retractable Module 

       In this section, we introduce the concept of P-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 P-small retractable if 𝑀 𝐻𝑜𝑚(𝑀, 𝐾)  ≠ 0 , 

for every non-zero P-small submodule 𝐾of 𝑀. Equivalently, 𝑀 is P-small retractable if there 

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

Remarks and Examples(4.2): 

1. It’s obvious that every P-small compressible module is P-small retractable 𝑚𝑜𝑑𝑢𝑙𝑒, but the 

converse is not true for instance𝑍4 is P-small retractable but not P-small compressible 𝑚𝑜𝑑𝑢𝑙𝑒 see 

remarks and examples (3.2) point(5).  

2. 𝑍 𝑎𝑠 𝑍 − 𝑚𝑜𝑑𝑢𝑙𝑒 is P-small retractable 𝑚𝑜𝑑𝑢𝑙𝑒, because it's P-small compressible 𝑚𝑜𝑑𝑢𝑙𝑒. 

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

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

4. Every retractable 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒 is P-small retractable 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒, but the converse is not 

true. 

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

6. Every compressible 𝑚𝑜𝑑𝑢𝑙𝑒 is P-small retractable 𝑚𝑜𝑑𝑢𝑙𝑒, but the converse is not true for 

instance 𝑍4 is P-small retractable but not P-small compressible 𝑚𝑜𝑑𝑢𝑙𝑒 see remarks and 

examples (3.2)point(5).   

7. A homomorphic image of a P-small retractable 𝑚𝑜𝑑𝑢𝑙𝑒 is a P-small retractable 𝑚𝑜𝑑𝑢𝑙𝑒. 

 

Remark(4.3):  The direct sum of P-small retractable 𝑚𝑜𝑑𝑢𝑙𝑒 is P-small retractable 𝑚𝑜𝑑𝑢𝑙𝑒.   

Proposition(4.4): A P-small submodule of  P-small retractable 𝑚𝑜𝑑𝑢𝑙𝑒 is also P-small retractable 

𝑚𝑜𝑑𝑢𝑙𝑒. 



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Proof: Let 0 ≠ 𝐾 ≪𝑃 𝑀 and 𝑀 be P-small retractable 𝑚𝑜𝑑𝑢𝑙𝑒 and let  0 ≠ 𝐿 ≤ 𝐾 ≪𝑃  𝑀 , by 

remarks and examples (1.2)point(3), [3]. 𝐿 ≪𝑃 𝑀 . Since 𝑀 is P-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 P-small retractable 𝑚𝑜𝑑𝑢𝑙𝑒. 

Proposition(4.5): Let 𝑀1 and 𝑀2 be isomorphic 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒𝑠. Then 𝑀1is P-small retractable if 

and only if 𝑀2is P-small retractable   

Proof: Let 0 ≠ 𝑁 ≪𝑃 𝑀1 and suppose that 𝑀2 is P-small retractable. Let 𝑓: 𝑀1 ⟶ 𝑀2 be an 

isomorphism. Then by[3] 0 ≠ 𝑓(𝑁) ≪𝑃 𝑀2. Put 𝐾 = 𝑓(𝑁) ≪𝑃 𝑀2, we get ℎ: 𝑀2 ⟶ 𝐾 is a 

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

 𝑓 −1(𝑓(𝑁)) = 𝑁. Hence we have a composition  

Η = 𝑔 ∘ ℎ ∘ 𝑓. Hence Η: 𝑀1 ⟶ 𝑁 is a monomorphism. Therefore 𝑀1is P-small retractable 𝑚𝑜𝑑𝑢𝑙𝑒. 

 

 

Proposition(4.7): Let 𝑀 be 𝑃𝑆 − ℎ𝑜𝑙𝑙𝑜𝑤 𝑚𝑜𝑑𝑢𝑙𝑒, then the following are equivalent  

(1)  𝑀 is retractable 𝑚𝑜𝑑𝑢𝑙𝑒. 

(2) 𝑀 is P-small retractable 𝑚𝑜𝑑𝑢𝑙𝑒. 

Proposition(4.8): If 𝑀 is P-small quasi-Dedekind 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒, then 𝑀 can't be  P-small 

retractable . 

Proof: Suppose that 𝑀 is  P-small quasi-Dedekind 𝑚𝑜𝑑𝑢𝑙𝑒 and let 𝑁 = 𝐾𝑒𝑟𝑓 ≤ 𝑀, but 𝑀 is P-

small quasi-Dedekind, then 𝐾𝑒𝑟𝑓 ≪𝑃 𝑀,  𝑓 ≠ 0 , thus 𝐻𝑜𝑚(𝑀, 𝐾𝑒𝑟𝑓) = 0. Therefore 𝑀 can't be  

P-small retractable 𝑚𝑜𝑑𝑢𝑙𝑒.  

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

for each 𝑓 ∈ 𝐻𝑜𝑚𝑅 (𝑁, 𝑀), 𝑓 ≠ 0  implies 𝐾𝑒𝑟𝑓 = 0, [5]. 

Definition(4.9): An 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒 𝑀 is called P-small monoform if for each non-zero submodule  

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

Remark(4.10): Every P-small compressible 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒 is P-small monoform, but not 

conversely. For example , 𝑍8 as 𝑍 − 𝑚𝑜𝑑𝑢𝑙𝑒 is P-small monoform but not P-small compressible.  

Proposition(4.11): Let 𝑀 be a quasi-Dedekind 𝑅 − 𝑚𝑜𝑑𝑢𝑙𝑒. Then 𝑀 is P-small monoform if and 

only if 𝑀 is P-small compressible.  

Proof: Suppose that 𝑀 is P-small monoform. Let 0 ≠ 𝑁 ≪𝑃  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  P-small compressible. Conversely it is clear by remark (4.10). 

  

5. Conclusion 

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

concepts called P-small compressible  and P-small retractable modules. Several characteristics of 

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

compressible and retractable are discuss 

Also we see  relations between P-small compressible modules and other related modules as P-

small retractable module P-small quasi-Dedekind, P-small monoform. 

 

 

 

 



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