Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (1) 2021 
 

116  
 

 

  

 

 

Nearly Primary-2-Absorbing Submodules and other Related Concepts 

 

     Omar A. Abdulla                             Ali Sh. Ajeel
 
                       Haibat  K. Mohammadali 

 

 

 

 

 

 

 
        

omar.aldoori87@gmail.com          Ali.shebl@st.tu.edu.iq
 
  

         
           H.mohammadali@tu.edu.iq 

 

  

 

 

  Abstract 

         Our aim in this paper is to introduce the notation of a nearly primary-2-absorbing 

submodule as generalization of 2-absorbing submodule where a proper submodule 𝐾 of an 𝑅-

module ∁ is called nearly primary-2-absorbing submodule if whenever 𝑎𝑏𝑥 ∈ 𝐾, for 𝑎, 𝑏, ∈ 𝑅, 

𝑥 ∈ ∁ implies that either 𝑎𝑥 ∈ 𝑟𝑎𝑑∁(𝐾) + 𝐽(∁) or 𝑏𝑥 ∈ 𝑟𝑎𝑑∁(𝐾) + 𝐽(∁) or 𝑎𝑏 ∈ [𝐾 +

𝐽(∁):𝑅 ∁]. We got many basic properties, examples and characterizations of this concept. 

Furthermore, characterizations of nearly primary-2-absorbing submodules in some classes of 

modules are inserted. Moreover, the behavior of nearly primary-2-absorbing submodule under 

𝑅-epimorphism is studied. 

 

Keywords: Primary submodules, prime submodules, multiplication modules, Artirian ring, 

projective 𝑅-modules , Jacobin of a modules . 

 

1. Introduction  

        Throughout this paper, all rings are commutative with identity and we assume all 𝑅-

modules are left unitary. Prime submodulse are  among the most famous concepts of modules 

theory, where a proper submodule 𝐻 of an 𝑅-module ∁ is said to be a prime submodule if 

whenever 𝑎𝑥 ∈ 𝐻, where 𝑎 ∈ 𝑅, 𝑥 ∈ ∁ implies that either 𝑥 ∈ 𝐻 or 𝑎∁ ⊆ 𝐻[1]. In addition, 

primary submodules is introduced in [2] as a generalization of a prime submodule, where a 

proper submodule 𝐻 of an 𝑅-module ∁ is said to be a primary submodule if whenever 𝑎𝑥 ∈ 𝐻, 

for 𝑎 ∈ 𝑅, 𝑥 ∈ ∁ implies that either 𝑥 ∈ 𝐻 or 𝑎𝑛∁ ⊆ 𝐻 for some 𝑛 ∈ 𝑍+. The well-known 

generalization of prime submodules is the concept of 2-absorbing submodule, where a proper 

Ibn Al Haitham Journal for Pure and Applied Science 

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

 

Doi: 10.30526/34.1.2560 

 

Article history: Received 4, Februaryber,2020, Accepted, 13, February 2020, Published in January 2021 

 

Directorate General of 

Education Salahaddin, the 

Ministry of Education, 

Tikrit, Iraq. 

Directorate General of 

Education Salahaddin, the 

Ministry of Education, Tikrit, 

Iraq. 

Department of Mathematics , 

College of Computer Sciences 

and Mathematics , Tikrit 

University, Tikrit, Iraq. 

 

mailto:omar.aldoori87@gmail.com
mailto:Ali.shebl@st.tu.edu.iq
mailto:H.mohammadali@tu.edu.iq


  

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Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (1) 2021 

 

submodule 𝐻 of an 𝑅-module ∁ is called 2-absorbing submodule if whenever 𝑎𝑏𝑥 ∈ 𝐻, for 

𝑎,𝑏 ∈ 𝑅, 𝑥 ∈ ∁ implies that either 𝑎𝑥 ∈ 𝐻 or 𝑏𝑥 ∈ 𝐻 or 𝑎𝑏∁ ⊆ 𝐻[3] is studied extensively. 

There are many generalizations of the concept of 2-absorbing, for example see [4 … 6]. Dubey 

in [7] introduced the concept of 2-absorbing primary submodule, where a proper submodule 𝐻 

of an 𝑅-module ∁ is called 2-absorbing primary submodule if whenever 𝑎𝑏𝑥 ∈ 𝐻, for 𝑎,𝑏 ∈ 𝑅, 

𝑥 ∈ ∁ implies that either 𝑎𝑥 ∈ 𝑟𝑎𝑑∁(𝐻) or 𝑏𝑥 ∈ 𝑟𝑎𝑑∁(𝐻) or 𝑎𝑏 ∈ [𝐻:𝑅 ∁], where 𝑟𝑎𝑑∁(𝐻) is 

the intersection of all prime submodule of ∁ containing 𝐻. Recall that an 𝑅-module ∁ is 

multiplication if every submodule 𝐻 of ∁ is of the form 𝐻 = 𝐽∁ for some ideal 𝐽 of 𝑅 [8]. 

Again recall that an 𝑅-module ∁ is said to be faithful if 𝑎𝑛𝑛𝑅 (∁) = (0). And a ring 𝑅 is said to 

be Artinian if 𝑅 satisfies D.CC on ideals of  𝑅, that is 𝐼1 ⊇ 𝐼2 ⊇ ⋯ ⊇ ⋯, then there exists 

𝑛 ∈ 𝑍+ such that 𝐼𝑛 = 𝐼𝑚 for some 𝑛 > 𝑚 [11]. Finally a ring 𝑅 is called a good ring, if 

𝐽(𝑅). ∁= 𝐽(∁) where ∁ is an 𝑅-module [12].  

 

2. Nearly Primary-2-Absorbing Submodules  

         In this paper, we will introduce the concept of nearly primary-2-absorbing submodule and 

give some basic results of these classes of submodules. And discuss on the relationships with 

class of 2-absorbing submodules  and nearly primary-2-absorbing submodules. 

Definition 1 

     A proper submodule 𝐻 of an 𝑅-module ∁ is said to be nearly primary-2-absorbing 

submodule of ∁, if whenever 𝑎𝑏𝑥 ∈ 𝐻, for 𝑎, 𝑏 ∈ 𝑅, 𝑥 ∈ ∁ implies that either 𝑎𝑥 ∈ 𝑟𝑎𝑑∁(𝐻) +

𝐽(∁) or 𝑏𝑥 ∈ 𝑟𝑎𝑑∁(𝐻) + 𝐽(∁) or 𝑎𝑏 ∈ [𝐻 + 𝐽(∁):𝑅 ∁]. And a proper ideal 𝐽 of a ring 𝑅 is called 

nearly primary-2-absorbing ideal of 𝑅, if 𝐽 is nearly primary-2-absorbing submodules of an 𝑅-

module 𝑅. 

 

Remarks and Examples 2 

1. It is clear that every 2-absorbing submodule of an 𝑅-module ∁ is a nearly primary-2-

absorbing submodule, while the reverse does not hold in general, the following example 

show that: Let ∁= 𝑍16, 𝑅 = 𝑍 and 𝐾 = 〈8̅〉. 𝐾 is not 2-absorbing submodule since 

2.2. 2̅ ∈ 𝐾 where 2 ∈ 𝑅 = 𝑍, 2̅ ∈ 𝑍16, then 2. 2̅ = 4̅ ∉ 𝐾 and 2.2 = 4 ∉ [𝐾:𝑅 𝑍16] = 8𝑍. 

But 𝐾 is a nearly primary-2-absorbing submodule of 𝑍16, since 𝐽(𝑍16) = 〈2̅〉 and 

𝑟𝑎𝑑∁(𝐾) = 〈2̅〉  for all  𝑎, 𝑏 ∈ 𝑍, 𝑥 ∈ 𝑍16 with 𝑎𝑏𝑥 ∈ 𝐾 = 〈8̅〉 implies that either 𝑎𝑥 ∈

𝑟𝑎𝑑∁(𝐾) + 𝐽(𝑍16) = 〈2̅〉 + 〈2̅〉 = 〈2̅〉  or 𝑏𝑥 ∈ 𝑟𝑎𝑑∁(𝐾) + 𝐽(𝑍16) = 〈2̅〉 or 𝑎𝑏 ∈

[〈8̅〉 + 𝐽(𝑍16):𝑍 𝑍16] = [〈2̅〉: 𝑍16] = 2𝑍. That is 2.2. 2̅ ∈ 𝐾, implies that 2. 2̅ = 4̅ ∈ 

〈2̅〉 + 〈2̅〉 = 〈2̅〉  or 2.2 = 4 ∈ [〈8̅〉 + 〈2̅〉:𝑍 𝑍16] = 2𝑍. 

2. It is clear that every prime submodule of an 𝑅-module ∁ is a nearly primary-2-absorbing 

submodule, while the reverse does not hold in general, the following example show that: 

Let ∁= 𝑍8, 𝑅 = 𝑍 and 𝐾 = 〈4̅〉 . 𝐾 is not prime submodule of 𝑍8, since 2. 2̅ ∈ 𝐾, where 

2 ∈ 𝑍, 2̅ ∈ 𝑍8 implies that 2̅ ∉ 𝐾 and 2 ∉ [𝐾:𝑍 ∁] = 4𝑍. But 𝐾 is a nearly primary-2-

absorbing submodule of ∁, since 2.2. 1̅ ∈ 𝐾, where 2 ∈ 𝑍, 1̅ ∈ 𝑍8 2 implies that 2. 1̅ ∈ 

𝑟𝑎𝑑∁(𝐾) + 𝐽(𝑍8) = 〈2〉  or 2.2 = 4 ∈ [𝐾 + 𝐽(∁):𝑍 ∁] = 2𝑍. 

3. It is clear that every primary submodule of an 𝑅-module ∁ is a nearly primary-2-absorbing 

submodule, while the reverse does not hold in general. The following example show that: 

Let ∁= 𝑍6, 𝑅 = 𝑍 and 𝐾 = 〈0̅〉 is a submodule of ∁. 𝐾 is a nearly primary-2-absorbing 

submodule of ∁ but not primary submodule, since 3 ∈ 𝑍, 2̅ ∈ 𝑍6 such that 3. 2̅ ∈ 𝐾, but 



  

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2̅ ∉ 𝐾 = 〈0̅〉 and 3 ∉ √[〈0̅〉: 𝑍6] = √6𝑍 = 6𝑍. But 𝐾 is a nearly primary-2-absorbing 

submodule of ∁= 𝑍6. For all 𝑎,𝑏 ∈ 𝑅,𝑥 ∈ ∁, with 𝑎𝑏𝑥 ∈ 𝐾 implies that either 𝑎𝑥 ∈

𝑟𝑎𝑑∁(𝐾) + 𝐽(∁) = 〈0̅〉 + 〈0〉 = 〈0̅〉  or 𝑏𝑥 ∈ 𝑟𝑎𝑑∁(𝐾) + 𝐽(∁) = 〈0̅〉 + 〈0〉 = 〈0̅〉 or 

𝑎𝑏 ∈ [〈0̅〉 + 〈0̅〉:𝑍 𝑍6] = 6𝑍. That is 2.3. 1̅ ∈ 𝐾,where 2,3 ∈ 𝑅, 1̅ ∈ 𝑍6 implies that 

2. 1̅ = 2̅ ∉ 〈0̅〉  but 2.3 = 6 ∈ [〈0̅〉 + 〈0̅〉:𝑍 𝑍6] = 6𝑍. 

  

The following results are characterizations of a nearly primary-2-absorbing submodules. 

 Proposition 3 

        Let ∁ be an 𝑅-module and 𝐾 is a proper submodule of ∁. Then 𝐾 is a nearly primary-2-

absorbing submodule of ∁ if and only if for each 𝑟, 𝑠 ∈ 𝑅 with 𝑟𝑠 ∉ [𝐾 + 𝐽(∁):𝑅 ∁], [𝐾:∁ 𝑟𝑠] ⊆

[𝑟𝑎𝑑∁(𝐾):∁ 𝑟] ∪ [𝑟𝑎𝑑∁(𝐾):∁ 𝑠]. 

Proof: 

 (⇒) Let 𝑥 ∈ [𝐾:∁ 𝑟𝑠], where 𝑟, 𝑠 ∈ 𝑅 and 𝑟𝑠 ∉ [𝐾 + 𝐽(∁):𝑅 ∁], implies that 𝑟𝑠𝑥 ∈ 𝐾. But 𝐾 is 

a nearly primary-2-absorbing submodule of ∁, and 𝑟𝑠 ∉ [𝐾 + 𝐽(∁):𝑅 ∁], then 𝑟𝑥 ∈ 𝑟𝑎𝑑∁(𝐾) +

𝐽(∁) or 𝑠𝑥 ∈ 𝑟𝑎𝑑∁(𝐾) + 𝐽(∁). That is either 𝑥 ∈ [𝑟𝑎𝑑∁(𝐾) + 𝐽(∁):∁ 𝑟] or 𝑥 ∈ [𝑟𝑎𝑑∁(𝐾) +

𝐽(∁):∁ 𝑠], thus 𝑥 ∈ [𝑟𝑎𝑑∁(𝐾) + 𝐽(∁):∁ 𝑟] ∪ [𝑟𝑎𝑑∁(𝐾) + 𝑠𝐽(∁):∁ 𝑠].  

Hence, [𝐾:∁ 𝑟𝑠] ⊆ [𝑟𝑎𝑑∁(𝐾):∁ 𝑟] ∪ [𝑟𝑎𝑑∁(𝐾):∁ 𝑠]. 

(⇐) Let 𝑟𝑠𝑥 ∈ 𝐾, where 𝑥 ∈ ∁ and 𝑟, 𝑠 ∈ 𝑅 with 𝑟𝑠 ∉ [𝐾 + 𝐽(∁):𝑅 ∁]. It follows that 𝑥 ∈

[𝐾:∁ 𝑟𝑠], by hypothesis 𝑥 ∈ [𝑟𝑎𝑑∁(𝐾):∁ 𝑟] ∪ [𝑟𝑎𝑑∁(𝐾):∁ 𝑠]. Hence 𝑥 ∈ [𝑟𝑎𝑑∁(𝐾):∁ 𝑟] or 

𝑥 ∈ [𝑟𝑎𝑑∁(𝐾):∁ 𝑠]. Therefore 𝑟𝑥 ∈ 𝑟𝑎𝑑∁(𝐾) + 𝐽(∁)  or 𝑠𝑥 ∈ 𝑟𝑎𝑑∁(𝐾) + 𝐽(∁) , that is 𝐾 is a 

nearly primary-2-absorbing submodule of ∁. 

Proposition 4 

       Let ∁ be an 𝑅-module and 𝐿 be a proper submodule of  ∁ . Then 𝐿 is a nearly primary-2-

absorbing submodule of  ∁  if and only if 𝑎𝑏𝐾 ⊆ 𝐿 for 𝑎, 𝑏 ∈ 𝑅 and 𝐾 is a submodule of  ∁ , 

with 𝑎𝑏 ∉ [𝐿 + 𝐽( ∁ ):𝑅  ∁ ], implies that 𝑎𝐾 ⊆ 𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ) or 𝑏𝐾 ⊆ 𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ). 

Proof: 

(⇒) Let 𝐿 be a nearly primary-2-absorbing submodule of ∁ , 𝑎𝑏𝐾 ⊆ 𝐿 with   𝑟, 𝑠 ∈ 𝑅 and 𝐾 is 

a submodule of ∁ with 𝑎𝑏 ∉ [𝐿 + 𝐽( ∁ ):𝑅  ∁]. Assume that 𝑎𝐾 ⊈ 𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ) and 

𝑏𝐾 ⊈ 𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ), then 𝑎𝑘1 ∉ 𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ) and 𝑏𝑘2 ∉ 𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ) for some 

𝑘1, 𝑘2 ∈ 𝐾. Now we have 𝑎𝑏𝑘1 ∈ 𝐿 but 𝐿 is a nearly primary-2-absorbing submodule of ∁ and 

𝑎𝑏 ∉ [𝐿 + 𝐽( ∁ ):𝑅  ∁] and 𝑎𝑘1 ∉ 𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ), then 𝑏𝑘1 ∈ 𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ). Also, since 

𝑎𝑏𝑘2 ∈ 𝐿 and 𝑎𝑏 ∉ [𝐿 + 𝐽( ∁ ):𝑅  ∁] and 𝑏𝑘2 ∉ 𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ), then 𝑎𝑘2 ∈ 𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ). 

Again since 𝑎𝑏(𝑘1 + 𝑘2) ∈ 𝐿 and 𝑎𝑏 ∉ [𝐿 + 𝐽( ∁ ):𝑅  ∁] we have 𝑎(𝑘1 + 𝑘2) ∈  𝑟𝑎𝑑∁(𝐿) +

𝐽( ∁ ) or 𝑏(𝑘1 + 𝑘2) ∈  𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ). Suppose that 𝑎(𝑘1 + 𝑘2) = 𝑎𝑘1 + 𝑎𝑘2 ∈ 𝑟𝑎𝑑∁(𝐿) +

𝐽( ∁ ), but 𝑎𝑘2 ∈ 𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ), it follows that 𝑎𝑘1 ∈ 𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ) a contradiction. 

Suppose that 𝑏(𝑘1 + 𝑘2) = 𝑏𝑘1 + 𝑏𝑘2 ∈  𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ), but 𝑏𝑘1 ∈ 𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ), we 

have 𝑏𝑘2 ∈ 𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ) a contradiction. Hence 𝑎𝐾 ⊆ 𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ) or 𝑏𝐾 ⊆

𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ). 

(⇐) Let  𝑎𝑏𝑥 ∈ 𝐿, where 𝑥 ∈ ∁ and 𝑎, 𝑏 ∈ 𝑅 with 𝑎𝑏 ∉ [𝐿 + 𝐽( ∁ ):𝑅  ∁ ]. So that  𝑎𝑏(𝑥) ⊆ 𝐿, 

it follows by hypothesis 𝑎(𝑥) ⊆  𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ) or 𝑏(𝑥) ⊆  𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ). That is 

𝑎𝑥 ∈  𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ) or 𝑏𝑥 ∈  𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ). Hence 𝐿 is a nearly primary-2-absorbing 

submodule of ∁.  

 



  

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Proposition 5 

       Let ∁ be an 𝑅-module and 𝐿 be a proper submodule of ∁. Then 𝐿 is a nearly primary-2-

absorbing submodule of ∁ if and only if 𝐼𝐽𝐾 ⊆ 𝐿, where 𝐼, 𝐽 are ideals of 𝑅 and 𝐾 is a 

submodule of ∁, implies that either 𝐼𝐽 ⊆ [𝐿 + 𝐽(∁):𝑅 ∁] or 𝐼𝐾 ⊆ 𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ) or 𝐽𝐾 ⊆

𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ). 

Proof: 

(⇐)   Clear. 

(⇒) Assume  that 𝐿 is a nearly primary-2-absorbing submodule of ∁, and 𝐼𝐽𝐾 ⊆ 𝐿, where 𝐼, 𝐽 

are ideals of 𝑅 and 𝐾 is a submodule of ∁ and 𝐼𝐽 ⊈ [𝐿 + 𝐽(∁):𝑅 ∁]. We must prove that 

𝐼𝐾 ⊆ 𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ) or 𝐽𝐾 ⊆ 𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ). Suppose that 𝐼𝐾 ⊈ 𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ )and 

𝐽𝐾 ⊈ 𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ). It follows that there exists 𝑟1 ∈ 𝐼 and 𝑟2 ∈ 𝐽 such that 𝑟1𝐾 ⊈ 𝑟𝑎𝑑∁(𝐿) +

𝐽( ∁ ) and 𝑟2𝐾 ⊈ 𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ). Now 𝑟1𝑟2𝐾 ⊆ 𝐿 with 𝑟1𝐾 ⊈ 𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ) and 𝑟2𝐾 ⊈

𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ) and 𝐿 is a nearly primary-2-absorbing submodule of  ∁ , impling that by 

Proposition(4) 𝑟1𝑟2 ∈ [𝐿 + 𝐽(∁):𝑅 ∁] . Since 𝐼𝐽 ⊈ [𝐿 + 𝐽(∁):𝑅 ∁]. It follows that there exists 

𝑠1 ∈ 𝐼, 𝑠2 ∈ 𝐽  such that 𝑠1𝑠2 ∉ [𝐿 + 𝐽(∁):𝑅 ∁] . Since 𝑠1𝑠2𝐾 ⊆ 𝐿, and  𝑠1𝑠2 ∉ [𝐿 + 𝐽(∁):𝑅 ∁] , 

we have by Proposition(4) either 𝑠1𝐾 ⊆ 𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ )  or 𝑠2𝐾 ⊆ 𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ) .  

Now, we discuss the following cases: 

Case one: Suppose that 𝑠1𝐾 ⊆ 𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ) but 𝑠2𝐾 ⊈ 𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ) Since 𝑟1𝑠2𝐾 ⊆ 𝐿 

and 𝐿 is a nearly primary-2-absorbing submodule of ∁ with 𝑠2𝐾 ⊈ 𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ) and 

𝑟1𝐾 ⊈ 𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ), implies that 𝑟1𝑠2 ∈ [𝐿 + 𝐽(∁):𝑅 ∁]  by Proposition(4). Also since 

𝑠1𝐾 ⊆ 𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ) but 𝑟1𝐾 ⊈ 𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ), it follows that (𝑟1 + 𝑠1)𝐾 ⊈ 𝑟𝑎𝑑∁(𝐿) +

𝐽( ∁ ). Since (𝑟1 + 𝑠1)𝑠2𝐾 ⊆ 𝐿 and 𝑠2𝐾 ⊈ 𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ) and (𝑟1 + 𝑠1)𝐾 ⊈ 𝑟𝑎𝑑∁(𝐿) +

𝐽( ∁ ) implies that by Proposition(4) (𝑟1 + 𝑠1)𝑠2 ∈ [𝐿 + 𝐽(∁):𝑅 ∁] . That is (𝑟1 + 𝑠1)𝑠2 = 𝑟1𝑠2 +

𝑠1𝑠2 ∈ [𝐿 + 𝐽(∁):𝑅 ∁]  and 𝑟1𝑠2 ∈ [𝐿 + 𝐽(∁):𝑅 ∁], implies that 𝑠1𝑠2 ∈ [𝐿 + 𝐽(∁):𝑅 ∁]  

 a contradiction. 

Case two: Let it be 𝑠2𝐾 ⊆ 𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ )  but 𝑠1𝐾 ⊈ 𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ) in similarly steps of 

Case one we get a contradiction. 

Case three: Assume that 𝑠1𝐾 ⊆ 𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ )  but 𝑠2𝐾 ⊆ 𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ) . Now since 

𝑠2𝐾 ⊆ 𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ) and 𝑟2𝐾 ⊈ 𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ), it follows that (𝑟2 + 𝑠2)𝐾 ⊈ 𝑟𝑎𝑑∁(𝐿) +

𝐽( ∁ ). We have 𝑟1(𝑟2 + 𝑠2)𝐾 ⊆ 𝐿 and 𝑟1𝐾 ⊈ 𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ) and (𝑟2 + 𝑠2)𝐾 ⊈ 𝑟𝑎𝑑∁(𝐿) +

𝐽( ∁ ),  by proposition(4) 𝑟1(𝑟2 + 𝑠2) = 𝑟1𝑟2 + 𝑟1𝑠2 ∈ [𝐿 + 𝐽(∁):𝑅 ∁].But 𝑟1𝑟2 ∈ [𝐿 + 𝐽(∁):𝑅 ∁] 

and 𝑟1𝑟2 + 𝑟1𝑠2 ∈ [𝐿 + 𝐽(∁):𝑅 ∁]. It follows that  𝑟1𝑠2 ∈ [𝐿 + 𝐽(∁):𝑅 ∁]. Now, since 𝑠1𝐾 ⊆

𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ) and 𝑟1𝐾 ⊈ 𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ), implies that (𝑟1 + 𝑠1)𝐾 ⊈ 𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ) 

since (𝑟1 + 𝑠1)𝑟2𝐾 ⊆ 𝐿 and 𝑟2𝐾 ⊈ 𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ) and (𝑟1 + 𝑠1)𝐾 ⊈ 𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ), it 

follows that (𝑟1 + 𝑠1)𝑟2 = 𝑟1𝑟2 + 𝑠1𝑟2 ∈ [𝐿 + 𝐽(∁):𝑅 ∁] by Proposition(4). Now, since 𝑟1𝑟2 ∈

[𝐿 + 𝐽(∁):𝑅 ∁] and 𝑟1𝑟2 + 𝑠1𝑟2 ∈ [𝐿 + 𝐽(∁):𝑅 ∁], implies that 𝑠1𝑟2 ∈ [𝐿 + 𝐽(∁):𝑅 ∁] . Also , 

since (𝑟1 + 𝑠1)(𝑟2 + 𝑠2)𝐾 ⊆ 𝐿 and (𝑟1 + 𝑠1)𝐾 ⊈ 𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ) and (𝑟2 + 𝑠2)𝐾 ⊈

𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ), it follows that  (𝑟1 + 𝑠1)(𝑟2 + 𝑠2) = 𝑟1𝑟2 + 𝑟1𝑠2 + 𝑠1𝑟2 + 𝑠1𝑠2 ∈

[𝐿 + 𝐽(∁):𝑅 ∁] by Proposition(4). Again since 𝑟1𝑟2, 𝑟1𝑠2, 𝑠1𝑟2 ∈ [𝐿 + 𝐽(∁):𝑅 ∁], we get that 

𝑠1𝑠2 ∈ [𝐿 + 𝐽(∁):𝑅 ∁] a contradiction. Thus, we have either 𝐼𝐾 ⊆ 𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ) or 𝐽𝐾 ⊆

𝑟𝑎𝑑∁(𝐿) + 𝐽( ∁ ).  

Proposition 6 

      Let 𝐻 be a proper submodule of an 𝑅-module ∁, with 𝑟𝑎𝑑∁(𝐻) is a prime submodule of ∁. 

Then 𝐻 is a nearly primary-2-absorbing submodule of ∁.  



  

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Proof: 

     Suppose that 𝑎𝑏𝑥 ∈ 𝐻, where 𝑎, 𝑏 ∈ 𝑅, 𝑥 ∈ ∁ and 𝑏𝑥 ∉ 𝑟𝑎𝑑∁(𝐻) + 𝐽( ∁ ). Since𝐻 ⊆

𝑟𝑎𝑑∁(𝐻), then 𝑎(𝑏𝑥) ∈ 𝑟𝑎𝑑∁(𝐻), but 𝑟𝑎𝑑∁(𝐻) is a prime submodule of ∁, then 𝑎∁⊆

𝑟𝑎𝑑∁(𝐻) ⊆ 𝑟𝑎𝑑∁(𝐻) + 𝐽(∁). That is 𝑎𝑥 ∈  𝑟𝑎𝑑∁(𝐻) + 𝐽( ∁ ), for some 𝑥 ∈ ∁. Thus 𝐻 is a 

nearly primary-2-absorbing submodule of ∁. 

Before we introduce the next result, we need to recall the following Lemma. 

Lemma 7[10] 

       If 𝑅 is a good ring, ∁ is an 𝑅-module and 𝑁 is a submodule of ∁, then 𝐽(∁) ∩ 𝑁 = 𝐽(𝑁). 

Proposition 8 

       Let 𝑅 be a good ring, 𝐿 and 𝐾 are proper submodules of an 𝑅-module ∁ with 𝐿 ⊊ 𝐾 and 

𝐽(∁) ⊆ 𝐾. If 𝐿 is a nearly primary-2-absorbing submodule of ∁, then  𝐿 is a nearly primary-2-

absorbing submodule of 𝐾. 

Proof: 

       Let 𝑟𝑠𝑥 ∈ 𝐿, where 𝑟, 𝑠 ∈ 𝑅, 𝑥 ∈ 𝐾 ⊆ ∁. Since 𝐿 is a nearly primary-2-absorbing 

submodule of ∁, implies that either 𝑟𝑥 ∈ 𝑟𝑎𝑑∁(𝐿) + 𝐽(∁) or 𝑠𝑥 ∈ 𝑟𝑎𝑑∁(𝐿) + 𝐽(∁) or 𝑟𝑠∁⊆ 𝐿 +

𝐽(∁). That is either 𝑟𝑥 ∈ (𝑟𝑎𝑑∁(𝐿) + 𝐽(∁) ) ∩ 𝐾 or  𝑠𝑥 ∈ (𝑟𝑎𝑑∁(𝐿) + 𝐽(∁) ) ∩ 𝐾 or 𝑟𝑠∁ ⊆ (𝐿 +

𝐽(∁)) ∩ 𝐾. But by modular law we have (𝑟𝑎𝑑∁(𝐿) + 𝐽(∁) ) ∩ 𝐾 = (𝑟𝑎𝑑∁(𝐿) ∩ 𝐾) + ( 𝐽(∁) ∩

𝐾) = (𝑟𝑎𝑑∁(𝐿) ∩ 𝐾) + 𝐽(𝐾) by Lemma(7). Thus we have either 𝑟𝑥 ∈ (𝑟𝑎𝑑∁(𝐿) ∩ 𝐾) +

𝐽(𝐾) ⊆ 𝑟𝑎𝑑∁(𝐿) + 𝐽(𝐾) or 𝑠𝑥 ∈∈ (𝑟𝑎𝑑∁(𝐿) ∩ 𝐾) + 𝐽(𝐾) ⊆ 𝑟𝑎𝑑∁(𝐿) + 𝐽(𝐾)  or 𝑟𝑠∁ ⊆

(𝐿 ∩ 𝐾) + (𝐽(∁) ∩ 𝐾) = (𝐿 ∩ 𝐾) + 𝐽(𝐾) ⊆ 𝐿 + 𝐽(𝐾). Hence 𝐿 is a nearly primary-2-absorbing 

submodule of 𝐾. 

Recall that for any submodules 𝐿, 𝐾 of a multiplication 𝑅-module ∁ with 𝐿 = 𝐼∁, 𝐾 = 𝐽∁ for 

some ideals 𝐼 and 𝐽 of 𝑅. The product 𝐿𝐾 = 𝐼∁. 𝐽∁= 𝐼𝐽∁. That is 𝐿𝐾 = 𝐼𝐾, in particular 

𝐿∁= 𝐼∁∁= 𝐼∁= 𝐿. Also for any 𝑥 ∈ ∁ we have 𝐿𝑥 = 𝐼𝑥[13]. 

The following result gives a characterization of nearly primary-2-absorbing submodules in 

class of multiplication modules. 

Proposition 9 

      Let ∁ be a multiplication 𝑅-module and 𝐿 is a proper submodule of ∁. Then 𝐿 is a nearly 

primary-2-absorbing submodule of ∁ if and only if, whenever  𝐿1𝐿2𝐿3 ⊆ 𝐿 for 𝐿1, 𝐿2, 𝐿3 are 

submodules of ∁, impling that either 𝐿1𝐿3 ⊆ 𝑟𝑎𝑑∁(𝐿) + 𝐽(∁) or 𝐿2𝐿3 ⊆ 𝑟𝑎𝑑∁(𝐿) + 𝐽(∁) or 

𝐿1𝐿2𝑊 ⊆ 𝐿 + 𝐽(∁). 

Proof: 

(⇒) Let 𝐿 be a nearly primary-2-absorbing submodule of ∁ and 𝐿1𝐿2𝐿3 ⊆ 𝐿 for 𝐿1, 𝐿2, 𝐿3 are 

submodules of ∁, with 𝐿1𝐿2 ∁ ⊈ 𝐿 + 𝐽(∁). Since ∁ is a multiplication, then 𝐿1 = 𝐼1∁ and 

𝐿2 = 𝐼2∁ for some ideals 𝐼1, 𝐼2, 𝐼3 of 𝑅. Clearly 𝐼1𝐼2𝐿3 ⊆ 𝐿 and 𝐼1𝐼2 ⊈ [𝐿 + 𝐽(∁):𝑅 ∁]. Since 𝐿 

is a nearly primary-2-absorbing submodule of ∁, implies that either 𝐼1𝐿3 ⊆ 𝑟𝑎𝑑∁(𝐿) + 𝐽(∁) or 

𝐼2𝐿3 ⊆ 𝑟𝑎𝑑∁(𝐿) + 𝐽(∁), it follows that either 𝐿1𝐿3 ⊆ 𝑟𝑎𝑑∁(𝐿) + 𝐽(∁)  or 𝐿2𝐿3 ⊆ 𝑟𝑎𝑑∁(𝐿) +

𝐽(∁) 

(⇐) Assume that 𝐼1𝐼2𝐾 ⊆ 𝐿, where 𝐼1, 𝐼2 are ideals of 𝑅, and 𝐾 is a submodule of ∁. Since ∁ is 

a multiplication, then 𝐼1𝐼2𝐾 = 𝐿1𝐿2𝐾 ⊆ 𝐿, by hypothesis either 𝐿1𝐾 ⊆  𝑟𝑎𝑑∁(𝐿) + 𝐽(∁)   or 

𝐿2𝐾 ⊆  𝑟𝑎𝑑∁(𝐿) + 𝐽(∁)  or 𝐿1𝐿2 ⊆ [𝐿 + 𝐽(∁)   :𝑅 ∁   ]. That is either 𝐼1𝐾 ⊆  𝑟𝑎𝑑∁(𝐿) + 𝐽(∁)or 

𝐼2𝐾 ⊆  𝑟𝑎𝑑∁(𝐿) + 𝐽(∁) or 𝐼1𝐼2 ⊆ [𝐿 + 𝐽(∁)   :𝑅 ∁   ]. Then by Proposition (5) 𝐿 is a nearly 

primary-2-absorbing submodule of ∁. 



  

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Recall that an 𝑅-epimorphism 𝑓: ∁⟶ ∁̅  is called a small epimorphism if ker (𝑓) is a small 

submodule of ∁ [10]. 

Lemma 10 [10, Corollary (9.1.5) ] 

       Let ∁ and ∁̅  be an 𝑅-module and 𝑁 be a proper submodule of ∁. If  𝑓: ∁⟶ ∁̅  is an 𝑅-

homomorphism, then  𝑓(𝐽(∁)) ⊆ 𝐽(∁̅) and 𝐽(𝑁) ⊆ 𝐽(∁). If 𝑓 is a small epimorphism then  

𝑓(𝐽(∁)) = 𝐽(∁̅) = 𝐽(𝑓(∁)) and 𝐽(∁) = 𝑓 −1(𝐽(∁̅)). 

Lemma 11 [2] 

       Let 𝑓: ∁⟶ ∁̅  be an 𝑅-epimorphism and 𝐿 is a submodule of ∁̅ with ker(𝑓) ⊆ 𝐿, then 

𝑓(𝑟𝑎𝑑∁(𝐿))) = 𝑟𝑎𝑑∁̅(𝑓(𝐿)). 

Proposition 12 

     Let 𝑓: ∁⟶ ∁̅  be a small 𝑅-epimorphism and �̅� is a nearly primary-2-absorbing submodule 

of ∁̅. Then 𝑓 −1(�̅�) be a nearly  primary-2-absorbing submodule of ∁. 

Proof: 

      Let 𝑟𝑠𝑥 ∈ 𝑓 −1(�̅�), where 𝑟, 𝑠 ∈ 𝑅, 𝑥 ∈ ∁, impling that 𝑟𝑠𝑓(𝑥) ∈  �̅�. Since �̅� is a nearly 

primary-2-absorbing submodule of ∁̅. It follows that either 𝑟𝑓(𝑥) ∈  𝑟𝑎𝑑∁̅(�̅�) + 𝐽(∁̅) or 

𝑠𝑓(𝑥) ∈  𝑟𝑎𝑑∁̅(�̅�) + 𝐽(∁̅)  or 𝑠𝑟∁̅ ⊆ �̅� + 𝐽(∁̅). Thus by Lemma (10) and by Lemma (11),      

we have either 𝑟𝑥 ∈ 𝑓 −1(𝑟𝑎𝑑∁̅(�̅�) + 𝑓
−1(𝐽(∁̅)) = 𝑟𝑎𝑑∁(𝑓

−1(�̅�)) + 𝐽(∁) or 𝑠𝑥 ∈

𝑓 −1(𝑟𝑎𝑑∁̅(�̅�) + 𝑓
−1(𝐽(∁̅)) = 𝑟𝑎𝑑∁(𝑓

−1(�̅�)) + 𝐽(∁)   or 𝑟𝑠∁ ⊆ 𝑓 −1(𝐾) + 𝐽(∁). Hence 𝑓 −1(�̅�) 

is a nearly primary-2-absorbing submodule of ∁. 

 

Proposition 13 

     Let 𝑓: ∁⟶ ∁̅  be a small 𝑅-epimorphism and 𝐾 is a nearly primary-2-absorbing submodule 

of ∁  with ker (𝑓) ⊆ 𝐾 . Then 𝑓(𝐾) is a nearly primary-2-absorbing submodule of ∁̅. 

 Proof: 

       Let 𝑟𝑠�̅� ∈ 𝑓(𝐾), where 𝑟, 𝑠 ∈ 𝑅, �̅� ∈ ∁̅. Since 𝑓 is onto, then 𝑓(𝑥) = �̅�  for some𝑥 ∈ ∁. 

Thus 𝑟𝑠𝑓(𝑥) ∈ 𝑓(𝐾), implies that 𝑟𝑠𝑓(𝑥) = 𝑓(𝑘) for some 𝑘 ∈ 𝐾, it follows that 𝑓(𝑟𝑠𝑥 −

𝑘) = 𝑜, implies that 𝑟𝑠𝑥 − 𝑘 ∈ ker (𝑓) ⊆ 𝐾, then 𝑟𝑠𝑥 ∈ 𝐾. But  𝐾 is a nearly primary-2-

absorbing submodule of ∁, then either 𝑟𝑥 ∈ 𝑟𝑎𝑑∁(𝐾) + 𝐽(∁)    or  𝑠𝑥 ∈ 𝑟𝑎𝑑∁(𝐾) + 𝐽(∁) or 

𝑟𝑠∁⊆ 𝐾 + 𝐽(∁), it follows that by Lemma(11) either 𝑟𝑓(𝑥) ∈ 𝑓(𝑟𝑎𝑑∁(𝐾)) + 𝑓(𝐽(∁)) ⊆

𝑟𝑎𝑑∁̅(𝑓(𝐾)) + 𝑓(𝐽(∁)) or 𝑠𝑓(𝑥) ∈ 𝑓(𝑟𝑎𝑑∁(𝐾)) + 𝑓(𝐽(∁)) ⊆ 𝑟𝑎𝑑∁̅(𝑓(𝐾)) + 𝑓(𝐽(∁)) or 

𝑟𝑠𝑓(∁) ⊆ 𝑓(𝐾) + 𝑓(𝐽(∁)) ⊆ 𝑓(𝐾) + 𝐽(∁̅). Also, by Lemma (12) either 𝑟�̅� ∈ 𝑟𝑎𝑑∁̅(𝑓(𝐾)) +

𝐽(∁̅) or 𝑠�̅� ∈ 𝑟𝑎𝑑∁̅(𝑓(𝐾)) + 𝐽(∁̅) or 𝑟𝑠∁̅ ⊆ 𝑓(𝐾) + 𝐽(∁̅). Hence 𝑓(𝐾) is a nearly primary-2-

absorbing submodule of ∁̅. 

Lemma 14 [9, Theorem(2.12)] 

       Let 𝑅 be a commutative ring with identity, 𝐿 be a proper submodule of a multiplication    

𝑅-module ∁ and  𝐽 = [𝐿:𝑅 ∁]. Then 𝑟𝑎𝑑∁(𝐿) = √𝐽. ∁= √[𝐿:𝑅 ∁]. ∁. 

Lemma 15 [2, Theorem 1, (1)] 

       Let ∁ is a module over Artinian Ring 𝑅, then 𝐽(𝑅). ∁= 𝐽(∁). 

Proposition 16 

     Let ∁ be a multiplication module over an Artinian ring 𝑅 and 𝐿 be a proper submodule of ∁. 

Then 𝐿 is a nearly primary-2-absorbing submodule of ∁ if and only if [𝐿:𝑅 ∁] is a nearly 

primary-2-absorbing ideal of 𝑅. 

 



  

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Proof: 

(⇒) Let 𝑎𝑐𝐼 ⊆ [𝐿:𝑅 ∁] for 𝑎, 𝑐 ∈ 𝑅, 𝐼 is an ideal of 𝑅 with 𝑎𝑐 ∉ [[[𝐿:𝑅 ∁] + J(R):𝑅 R]] =

[𝐿:𝑅 ∁] + J(R), that is 𝑎𝑐∁⊈ [𝐿:𝑅 ∁]. ∁ + J(R). ∁. But ∁ is a module over Artinian ring, then by 

Lemma (15) 𝐽(𝑅). ∁= 𝐽(∁), that is 𝑎𝑐∁⊈ 𝐿 + J(∁) i.e. 𝑎𝑐 ∉ [𝐿 + J(∁):𝑅 ∁] . Now, since 

𝑎𝑐𝐼 ⊆ [𝐿:𝑅 ∁], then 𝑎𝑐(𝐼∁) ⊆ 𝐿. But 𝐿 is a nearly primary-2-absorbing submodule of ∁, then by 

Proposition (4) either 𝑎(𝐼∁) ⊆ 𝑟𝑎𝑑∁(𝐿) + 𝐽(∁) or 𝑐(𝐼∁) ⊆ 𝑟𝑎𝑑∁(𝐿) + 𝐽(∁). But by  

Lemma(14) 𝑟𝑎𝑑∁(𝐿) = √[𝐿:𝑅 ∁] . ∁ and by Lemma(15) 𝐽(𝑅). ∁= 𝐽(∁). Thus either 𝑎𝐼∁ ⊆

√[𝐿:𝑅 ∁]. ∁ + 𝐽(𝑅). ∁ or 𝑐𝐼∁ ⊆ √[𝐿:𝑅 ∁]. ∁ + 𝐽(𝑅). ∁, it follows that either 𝑎𝐼 ⊆ √[𝐿:𝑅 ∁]. ∁ +

𝐽(𝑅) or 𝑐𝐼 ⊆ √[𝐿:𝑅 ∁]. ∁ + 𝐽(𝑅). Therefore [𝐿:𝑅 ∁] is a nearly primary-2-absorbing ideal of 𝑅. 

(⇐) Assume that [𝐿:𝑅 ∁] is a nearly primary-2-absorbing ideal of 𝑅, and 𝑟𝑠𝐾 ⊆ 𝐿, for 𝑟,𝑠 ∈ 𝑅, 

𝐾 is a submodule of ∁ with 𝑟𝑠 ∉ [𝐿 + 𝐽(∁):𝑅 ∁], it follows that 𝑟𝑠∁ ⊈ 𝐿 + 𝐽(∁). But ∁ is a 

module over Artinian ring, then by lemma (15)  𝐽(𝑅). ∁= 𝐽(∁). Thus 𝑟𝑠∁ ⊈ [𝐿:𝑅 ∁]. ∁ + 𝐽(𝑅). ∁, 

it follows that 𝑟𝑠 ⊈ [𝐿:𝑅 ∁] + 𝐽(𝑅) = [[[𝐿:𝑅 ∁] + J(R):𝑅 R]]. Now, since 𝑟𝑠𝐾 ⊆ 𝐿 and ∁ is a 

multiplication, then 𝐾 = 𝐼∁  for some ideal 𝐼 of 𝑅. Hence 𝑟𝑠𝐼∁ ⊆ 𝐿, implies that 𝑟𝑠𝐼 ⊆ [𝐿:𝑅 ∁]. 

But [𝐿:𝑅 ∁] is a nearly primary-2-absorbing ideal of 𝑅 and 𝑟𝑠 ∉ [[[𝐿:𝑅 ∁] + J(R):𝑅 R]], then by 

proposition (4) we have either 𝑟𝐼 ⊆ √[𝐿:𝑅 ∁] + 𝐽(𝑅) or 𝑠𝐼 ⊆ √[𝐿:𝑅 ∁] + 𝐽(𝑅). That is 

𝑟𝐼∁ ⊆ √[𝐿:𝑅 ∁] . ∁ + 𝐽(𝑅). ∁  or 𝑠𝐼∁ ⊆ √[𝐿:𝑅 ∁] . ∁ + 𝐽(𝑅). ∁ . But ∁ is a module over Artinian 

ring, then by Lemma (15)  𝐽(𝑅). ∁= 𝐽(∁) and by Lemma(14) 𝑟𝑎𝑑∁(𝐿) = √[𝐿:𝑅 ∁] . ∁ we get 

either 𝑟𝐾 ⊆ 𝑟𝑎𝑑∁(𝐿) + 𝐽(∁) or 𝑠𝐾 ⊆ 𝑟𝑎𝑑∁(𝐿) + 𝐽(∁). Hence by Proposition (4) 𝐿 is a nearly 

primary-2-absorbing submodule of ∁.  

 

 Lemma 17 [10, proposition (17.10)] 

       If 𝑃 is projective 𝑅-module, then  𝐽(𝑅). 𝑃 = 𝐽(𝑃). 

Proposition 18 

     Let ∁ be a multiplication projective 𝑅-module and 𝐿 is a proper submodule of ∁. Then 𝐿 is a 

nearly primary-2-absorbing submodule of ∁ if and only if [𝐿:𝑅 ∁] is a nearly primary-2-

absorbing ideal of 𝑅. 

Proof: 

(⇒) Let 𝑎𝑐𝐼 ⊆ [𝐿:𝑅 ∁] for 𝑎, 𝑐 ∈ 𝑅, 𝐼 is an ideal of 𝑅 with 𝑎𝑐 ∉ [[[𝐿:𝑅 ∁] + J(R):𝑅 R]] =

[𝐿:𝑅 ∁] + J(R), that is 𝑎𝑐∁⊈ [𝐿:𝑅 ∁]. ∁ + J(R). ∁. But ∁ is projective 𝑅-module, then by Lemma 

(17) 𝐽(𝑅). ∁= 𝐽(∁), that is 𝑎𝑐∁⊈ 𝐿 + J(∁) i.e. 𝑎𝑐 ∉ [𝐿 + J(∁):𝑅 ∁] . Now, since 𝑎𝑐𝐼 ⊆ [𝐿:𝑅 ∁], 

then 𝑎𝑐(𝐼∁) ⊆ 𝐿. But 𝐿 is a nearly primary-2-absorbing submodule of ∁, then by proposition 

(4) either 𝑎(𝐼∁) ⊆ 𝑟𝑎𝑑∁(𝐿) + 𝐽(∁) or 𝑐(𝐼∁) ⊆ 𝑟𝑎𝑑∁(𝐿) + 𝐽(∁). But by  Lemma(14) 𝑟𝑎𝑑∁(𝐿) =

√[𝐿:𝑅 ∁] . ∁ and by Lemma(17) 𝐽(𝑅). ∁= 𝐽(∁). Thus either 𝑎𝐼∁ ⊆ √[𝐿:𝑅 ∁]. ∁ + 𝐽(𝑅). ∁ or 

𝑐𝐼∁ ⊆ √[𝐿:𝑅 ∁]. ∁ + 𝐽(𝑅). ∁. It follows that either 𝑎𝐼 ⊆ √[𝐿:𝑅 ∁]. ∁ + 𝐽(𝑅) or 𝑐𝐼 ⊆

√[𝐿:𝑅 ∁]. ∁ + 𝐽(𝑅). Therefore, [𝐿:𝑅 ∁] is a nearly primary-2-absorbing ideal of 𝑅. 

(⇐) Assume that [𝐿:𝑅 ∁] is a nearly primary-2-absorbing ideal of 𝑅, and 𝑟𝑠𝐾 ⊆ 𝐿, for 𝑟,𝑠 ∈ 𝑅, 

𝐾 is a submodule of ∁ with 𝑟𝑠 ∉ [𝐿 + 𝐽(∁):𝑅 ∁], it follows that 𝑟𝑠∁ ⊈ 𝐿 + 𝐽(∁). But ∁ is a 

project 𝑅-module, then by Lemma (17)  𝐽(𝑅). ∁= 𝐽(∁). Thus 𝑟𝑠∁ ⊈ [𝐿:𝑅 ∁]. ∁ + 𝐽(𝑅). ∁. It 

follows that 𝑟𝑠 ⊈ [𝐿:𝑅 ∁] + 𝐽(𝑅) = [[[𝐿:𝑅 ∁] + J(R):𝑅 R]]. Now, since 𝑟𝑠𝐾 ⊆ 𝐿 and ∁ is a 



  

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multiplication, then 𝐾 = 𝐼∁  for some ideal 𝐼 of 𝑅. Hence 𝑟𝑠𝐼∁ ⊆ 𝐿, implies that 𝑟𝑠𝐼 ⊆ [𝐿:𝑅 ∁]. 

But [𝐿:𝑅 ∁] is a nearly primary-2-absorbing ideal of 𝑅 and 𝑟𝑠 ∉ [[[𝐿:𝑅 ∁] + J(R):𝑅 R]], then by 

Proposition (4) we have either 𝑟𝐼 ⊆ √[𝐿:𝑅 ∁] + 𝐽(𝑅) or 𝑠𝐼 ⊆ √[𝐿:𝑅 ∁] + 𝐽(𝑅). That is 

𝑟𝐼∁ ⊆ √[𝐿:𝑅 ∁] . ∁ + 𝐽(𝑅). ∁  or 𝑠𝐼∁ ⊆ √[𝐿:𝑅 ∁] . ∁ + 𝐽(𝑅). ∁ . But ∁ is projective 𝑅-module, 

then by Lemma (17)  𝐽(𝑅). ∁= 𝐽(∁) and by Lemma(14) 𝑟𝑎𝑑∁(𝐿) = √[𝐿:𝑅 ∁] . ∁ we get either 

𝑟𝐾 ⊆ 𝑟𝑎𝑑∁(𝐿) + 𝐽(∁) or 𝑠𝐾 ⊆ 𝑟𝑎𝑑∁(𝐿) + 𝐽(∁). Hence by proposition (4) 𝐿 is a nearly 

primary-2-absorbing submodule of ∁.  

 

We need to invite the following finding before we study the next propositions.  

Lemma 19[14, Corollary of Theorem. 9] 

         Let 𝐼1 and 𝐼2 are ideals of a ring 𝑅 and ∁ is a finitely generated multiplication 𝑅-module. 

Then 𝐼1∁ ⊆ 𝐼2∁ if and only if 𝐼1 ⊆ 𝐼2 + 𝑎𝑛𝑛𝑅 (∁). 

Lemma 20[15, Proposition. (2.4)] 

      Let ∁ be a multiplication 𝑅-module and 𝐼 is an ideal of 𝑅 such that 𝑎𝑛𝑛𝑅 (∁) ⊆ 𝐼, then 

𝑟𝑎𝑑∁(𝐼∁) = √𝐼∁. 

Proposition 21 

     Let ∁ be a faithful finitely generated multiplication 𝑅-module over an Artinian ring 𝑅 and 𝐼 

is a nearly primary-2-absorbing ideal of 𝑅 and 𝐼∁≠ ∁. Then 𝐼∁ is a nearly primary-2-absorbing 

submodule of ∁.  

Proof:  

      Let 𝑎𝑐𝑥 ∈ 𝐼∁ for 𝑎,𝑐 ∈ 𝑅, 𝑥 ∈ ∁, then 𝑎𝑐(𝑥) ⊆ 𝐼∁, implies that 𝑎𝑏𝐽∁ ⊆ 𝐼∁ for some ideal 𝐽 

of 𝑅 since ∁ is a multiplication. Hence by Lemma(19) 𝑎𝑐𝐽 ⊆ 𝐼 + 𝑎𝑛𝑛𝑅 (∁), but ∁ is a faithful. It 

follows that 𝑎𝑛𝑛𝑅 (∁) = (0), that is 𝑎𝑐𝐽 ⊆ 𝐼. Since 𝐼 is a nearly primary-2-absorbing ideal of 𝑅, 

then by Proposition(4) either 𝑎𝐽 ⊆ √𝐼 + 𝐽(𝑅) or 𝑐𝐽 ⊆ √𝐼 + 𝐽(𝑅) or 𝑎𝑐 ∈ [𝐼 + 𝐽(𝑅): 𝑅] = 𝐼 +

𝐽(𝑅). It follows that 𝑎𝐽∁ ⊆ √𝐼∁ + 𝐽(𝑅)∁ or 𝑐𝐽∁ ⊆ √𝐼∁ + 𝐽(𝑅)∁  or 𝑎𝑏∁ ⊆ 𝐼∁ + 𝐽(𝑅)∁. But by 

Lemma(15) 𝐽(𝑅)∁= 𝐽(∁) and by Lemma(20) √𝐼∁ = 𝑟𝑎𝑑∁(𝐼∁). Thus either 𝑎𝑥 ∈ 𝑟𝑎𝑑∁(𝐼∁) +

𝐽(∁) or 𝑐𝑥 ∈ 𝑟𝑎𝑑∁(𝐼∁) + 𝐽(∁) or 𝑎𝑏∁⊆ 𝐼∁ + 𝐽(∁). Hence 𝐼∁ is a nearly primary-2-absorbing 

submodule of ∁. 

Proposition 22 

       Let ∁ be a faithful finitely generated multiplication 𝑅-module over an Artinian ring 𝑅 and 

𝐾 be a proper submodule of ∁. Then the next statements are equivalent. 

1. L is a nearly primary-2-absorbing submodule of ∁. 

2. [L:𝑅 ∁] is a nearly primary-2-absorbing ideal of 𝑅. 

3. L = J∁ for some nearly primary-2-absorbing ideal of 𝑅.  

Proof: 

1 ⇔ 2 Via Proposition(16). 

2 ⇒ 3 Since [L:𝑅 ∁] is a nearly primary-2-absorbing ideal of 𝑅 with 𝑎𝑛𝑛𝑅 (∁) = [0: ∁] ⊆

[L:𝑅 ∁] and L = [L:𝑅 ∁]∁, implies that L = I∁ where I = [L:𝑅 ∁] is a nearly primary-2-absorbing 

ideal of 𝑅. 

3 ⟹ 2 Suppose that 𝐿 = 𝐽∁ for some nearly primary-2-absorbing ideal 𝐽 of 𝑅. Since ∁ is 

multiplication, then L = [L:𝑅 ∁]∁= I∁. Since ∁ is a faithful finitely generated multiplication 𝑅-

module over an Artinian ring 𝑅, then we have [L:𝑅 ∁] = J. Thus [L:𝑅 ∁] is a nearly primary-2-

absorbing ideal of 𝑅.   



  

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3. Conclusion 

        In this article we introduce a new generalization of (prime, primary, 2-absorbing) 

submodules called a nearly primary-2-absorbing submodules and we explain the converse 

implication of the above by examples. Many characterizations of this generalization are 

introduced. Relationships of this generalization with other classes of modules are given. 

 

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