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Omar A. Abdulla                                                  Haibat K. Mohammadali 

                        

 

                                                       

 

 

 
 

Abstract 

      Let 𝑅 be a commutative ring with identity. The aim of this paper is introduce the notion of 

a pseudo primary-2-absorbing submodule as generalization of 2-absorbing submodule and a 

pseudo-2-absorbing submodules. A proper submodule 𝐾 of an 𝑅-module 𝑊 is called pseudo 

primary-2-absorbing if whenever 𝑟𝑠𝑥 ∈ 𝐾, for 𝑟, 𝑠 ∈ 𝑅, 𝑥 ∈ 𝑊, implies that either                 

𝑟𝑥 ∈  𝑊 − 𝑟𝑎𝑑(𝐾) + 𝑠𝑜𝑐(𝑊) or 𝑠𝑥 ∈ 𝑊 − 𝑟𝑎𝑑(𝐾) + 𝑠𝑜𝑐(𝑊) or 𝑟𝑠𝑊 ⊆ 𝐾 + 𝑠𝑜𝑐(𝑊). 

Many basic properties, examples and characterizations of these concepts are given. 

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

modules are introduced. Moreover, the behavior of a pseudo primary-2-absorbing submodule 

under 𝑅-homomorphism is studied.   

  

Keywords: Primary submodules, pseudo-2-absorbing submodules, pseudo primary-2-

absorbing submodules, multiplication modules, non-singular modules, socle of a modules. 
 

1. Introduction and Basic Concepts  

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

𝑅-modules are left unitary. Among the famous concepts of modules theory is prime 

submodules, where a proper submodule 𝐾 of an 𝑅-module 𝑊 is said to be a prime submodule 

if whenever 𝑟𝑥 ∈ 𝐾 where 𝑟 ∈ 𝑅, 𝑥 ∈ 𝑊, implies that either 𝑥 ∈ 𝑊 or 𝑟𝑊 ⊆ 𝐾[1]. Primary 

submodule was introduced in [2]. as a generalization of a prime submodule, where a proper 

submodule 𝐾 of an 𝑅-module 𝑊 is called a primary submodule if whenever 𝑟𝑥 ∈ 𝐾 for 𝑟 ∈

𝑅, 𝑥 ∈ 𝑊, implies that either 𝑥 ∈ 𝐾 or 𝑟𝑛𝑊 ⊆ 𝐾 for some 𝑛 ∈ 𝑍+. Recently many 

generalizations of prime submodules were introduced such as (app-prime, 𝜑-prime, Nearly-

prime) submodules see[3 − 5]. Darani and Soheilinia in[6]. introduced the concept of 2-

absorbing submodule as a generalization of prime submodule, where a proper submodule 𝐾 of 

an 𝑅-module 𝑊 is said to be 2-absorbing submodule if whenever 𝑟𝑠𝑥 ∈ 𝐾 for 𝑟, 𝑠 ∈ 𝑅, 

𝑥 ∈ 𝑊, implies that either 𝑟𝑥 ∈ 𝐾 or 𝑠𝑥 ∈ 𝐾 or 𝑟𝑠𝑊 ⊆ 𝐾. In recent decades several 

generalization of 2-absorbing submodules were introduced such as nearly 2-absorbing 

submodule, nearly quasi-2-absorbing submodule, pseudo-2-absorbing submodule and pseudo 

Ibn Al Haitham Journal for Pure and Applied Science 

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

 

Doi:10.30526/32.3.2290 

Pseudo Primary-2-Absorbing Submodules and Some Related Concepts 

 

Article history: Received 3 March 2019, Accepted 20 March 2019, Publish September 

2019 

 

Department of Mathematics, College of Computer Science and Mathematics, 

University of Tikrit, Iraq. 

 omar.aldoori87@gmail.com                      dr.mohammadali2013@gmail.com 

 

mailto:omar.aldoori87@gmail.com
mailto:dr.mohammadali2013@gmail.com


  

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quasi-2-absorbing submodule see[6 − 9]. Badwi et, in [10]. introduced the concept of 2-

absorbing primary ideal, where a proper ideal 𝐼 of a ring 𝑅 is called 2-absorbing primary, if 

whenever 𝑎𝑏𝑐 ∈ 𝐼 for 𝑎, 𝑏, 𝑐 ∈ 𝑅, implies that either 𝑎𝑏 ∈ 𝐼 or 𝑎𝑐 ∈ √𝐼 or 𝑏𝑐 ∈ √𝐼 where 

√𝐼 = {𝑟 ∈ 𝑅: 𝑟𝑛 ∈ 𝐼, for some n ∈ 𝑍+}. This led us to introduce the concept of a pseudo 

primary-2-absorbing submodule, which is generalization of 2-absorbing submodule and 

pseudo- 2-absorbing submodule. Many basic properties, characterization and examples of this 

concept are given. The residual of submodule 𝐾 is denoted by [𝐾: 𝑊] is an ideal of 𝑅 defined 

by {𝑟 ∈ 𝑅: 𝑟𝑊 ⊆ 𝐾}[1]. The  radical of a submodule 𝐾 of 𝑊 denoted by 𝑊 − 𝑟𝑎𝑑(𝐾) or 

𝑟𝑎𝑑𝑊(𝐾)  is defined to be the intersection of all prime submodule of 𝑊 containing 𝐾, if  𝑊 

has no prime submodules containing 𝐾, then we say 𝑊 − 𝑟𝑎𝑑(𝐾) = 𝑊 and 𝑊 ⊆ 𝑊 −

𝑟𝑎𝑑(𝐾)[2]. Socle of a module 𝑊 defined by the intersection of all essential submodules of 

𝑊, denoted by 𝑠𝑜𝑐(𝑊)[11]. Recall that an 𝑅-module 𝑊 is multiplication, if every submodule 

𝐿 of 𝑊 is of the form 𝐿 = 𝐼𝑊 for some ideal 𝐼 of a ring  [12]. Recall that an 𝑅-module 𝑊 is 

called faithful if 𝑎𝑛𝑛(𝑊) = (0). Recall that an 𝑅-module 𝑊 is called non-singular if 

𝑍(𝑊) = 𝑊 where (𝑊) = {𝑦 ∈ 𝑊: 𝑦𝐼 = (0), for some essential ideal I of  R} [11]. 
 

2. Pseudo Primary-2-Absorbing Submodules  

       In this section we define the concept of a pseudo primary-2-absorbing submodule and 

give some basic results of these types of submodules and discuss on the relationships with 

class of 2-absorbing submodules and pseudo-2-absorbing submodules. 

Definition (1) 

     A proper submodule 𝐾 of an 𝑅-module 𝑊 is said to be a pseudo primary-2-absorbing 

submodule of 𝑊, if whenever 𝑟𝑠𝑥 ∈ 𝐾, for 𝑟, 𝑠 ∈ 𝑅, 𝑥 ∈ 𝑊, implies that either 𝑟𝑥 ∈ 𝑊 −

𝑟𝑎𝑑(𝐾) + 𝑠𝑜𝑐(𝑊) or 𝑠𝑥 ∈ 𝑊 − 𝑟𝑎𝑑(𝐾) + 𝑠𝑜𝑐(𝑊) or 𝑟𝑠 ∈ [𝐾 + 𝑠𝑜𝑐(𝑊):𝑅 𝑊]. And a 

proper ideal 𝐼 of a ring 𝑅 is called a pseudo primary-2-absorbing ideal of 𝑅, if 𝐼 is  pseudo 

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 𝑊 is a pseudo primary-2-

absorbing submodule, while the converse is not true  in general, the following example 

shows that: Let 𝑊 = 𝑍12, 𝑅 = 𝑍 and 𝐾 = 〈0̅〉. 𝐾 is not 2-absorbing submodule since 

2.3. 2̅ ∈ 𝐾 where 2,3 ∈ 𝑍, 2̅ ∈ 𝑍12, then 2. 2̅ = 4̅ ∉ 𝐾 and 3. 2̅ = 6̅ ∉ 𝐾 and 2.3 = 6 ∉

[𝐾: 𝑍12] = 12𝑍. But 𝐾 is a pseudo primary-2-absorbing submodule of 𝑍12, since 

𝑠𝑜𝑐(𝑍12) = 〈2̅〉 and 𝑊 − 𝑟𝑎𝑑(𝐾) = 〈6̅〉  for all  𝑟, 𝑠 ∈ 𝑍, 𝑥 ∈ 𝑍12 with 𝑟𝑠𝑥 ∈ 〈0̅〉, 

implies that either 𝑟𝑥 ∈ 〈6̅〉 + 𝑠𝑜𝑐(𝑍12) = 〈2̅〉  or 𝑠𝑥 ∈ 〈6̅〉 + 𝑠𝑜𝑐(𝑍12) = 〈2̅〉 or 𝑟𝑠 ∈

[〈0̅〉 + 𝑠𝑜𝑐(𝑍12): 𝑍12] = [〈2̅〉: 𝑍12] = 2𝑍. That is 2.3. 2̅ ∈ 𝐾, implies that 2. 2̅ = 4̅ ∈ 

〈6̅〉 + 〈2̅〉 = 〈2̅〉  or 3. 2̅ = 6̅ ∈ 〈6̅〉 + 〈2̅〉 = 〈2̅〉 or 2.3 = 6 ∈ [〈0̅〉 + 〈2̅〉: 𝑍12] = 2𝑍. 

2. It is clear that every pseudo-2-absorbing submodule of an 𝑅-module 𝑊 is a pseudo 

primary-2-absorbing submodule, while the converse is not true  in general, the following 

example shows that: Let 𝑊 = 𝑍, 𝑅 = 𝑍 and 𝐾 = 8𝑍 where 𝐾 be a submodule of 𝑊. 𝐾 is 

not pseudo-2-absorbing submodule since 2.2.2 ∈ 8𝑍 but 2.2 ∉ 8𝑍 + 𝑠𝑜𝑐(𝑍) = 8𝑍 +

(0) = 8𝑍 and  2.2 = 4 ∉ [8𝑍 + 𝑠𝑜𝑐(𝑍): 𝑍] = 8𝑍. But 𝐾 is a pseudo primary-2-

absorbing submodule of 𝑊 since 2.2.2 ∈ 8𝑍, then 2.2 = 4 ∈ 𝑊 − 𝑟𝑎𝑑(8𝑍) + 𝑠𝑜𝑐(𝑍) =

2𝑍 + (0) = 2𝑍. That is for all  𝑟, 𝑠 ∈ 𝑅, 𝑥 ∈ 𝑊 with 𝑟𝑠𝑥 ∈ 𝐾, implies that either 

𝑟𝑥 ∈ 𝑊 − 𝑟𝑎𝑑(𝐾) + 𝑠𝑜𝑐(𝑊) = 2𝑍 or 𝑠𝑥 ∈ 2𝑍 or 𝑟𝑠 ∈ [8𝑍:𝑍 𝑍] = 8𝑍. 



  

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3. It is clear that every primary submodule of an 𝑅-module 𝑊 is a pseudo primary-2-

absorbing submodule, while the converse is not true  in general, the following example 

shows that: Let 𝑊 = 𝑍12, 𝑅 = 𝑍 and 𝐾 = 〈0̅〉 is a submodule of 𝑊. 𝐾 is a pseudo 

primary-2-absorbing submodule of 𝑊 but not primary submodule, since 3 ∈ 𝑍, 4̅ ∈ 𝑍12 

such that 3. 4̅ ∈ 𝐾, but 4̅ ∉ 𝐾 = 〈0̅〉 and 3 ∉ √[〈0̅〉: 𝑍12] = √12𝑍 = 6𝑍. 

4. It is clear that every prime submodule of an 𝑅-module 𝑊 is a pseudo primary-2-

absorbing submodule, while the converse is not true  in general, the following example 

shows that: Let 𝑊 = 𝑍, 𝑅 = 𝑍 and 𝐾 = 6𝑍 . 𝐾 is not prime submodule of 𝑊, since 2,3𝑍 

with  2.3 ∈ 𝐾, but 3 ∉ 𝐾 and 2 ∉ [𝐾:𝑍 𝑍] = 6𝑍. But 𝐾 is a pseudo primary-2-absorbing 

submodule of 𝑊, since 2,3,1 ∈ 𝑍 with 2.3.1 ∈ 𝐾, implies that 2.3 ∈ [𝐾 + 𝑠𝑜𝑐(𝑊): 𝑊] =

6𝑍 because 𝑠𝑜𝑐(𝑊) = (0). 2.1 ∉ 𝑊 − 𝑟𝑎𝑑(𝐾) + 𝑠𝑜𝑐(𝑊) = 6𝑍 and 3.1 ∉ 𝑊 −

𝑟𝑎𝑑(𝐾) + 𝑠𝑜𝑐(𝑊) = 6𝑍. That is for all 𝑟, 𝑠 ∈ 𝑅, 𝑥 ∈ 𝑊, with 𝑟𝑠𝑥 ∈ 𝐾, implies that 

either 𝑟𝑥 ∈ 𝑊 − 𝑟𝑎𝑑(𝐾) + 𝑠𝑜𝑐(𝑊) or 𝑠𝑥 ∈ 𝑊 − 𝑟𝑎𝑑(𝐾) + 𝑠𝑜𝑐(𝑊) or 𝑟𝑠 ∈

[𝐾 + 𝑠𝑜𝑐(𝑊):𝑅 𝑊].   

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

 Proposition (3) 

        Let 𝑊 be an 𝑅-module and 𝐾 is a proper submodule of 𝑊. Then 𝐾 is a pseudo primary-

2-absorbing submodule of 𝑊 if and only if for each 𝑟, 𝑠 ∈ 𝑅 with 𝑟𝑠 ∉ [𝐾 + 𝑠𝑜𝑐(𝑊):𝑅 𝑊], 

[𝐾:𝑊 𝑟𝑠] ⊆ [𝑊 − 𝑟𝑎𝑑(𝐾) + 𝑠𝑜𝑐(𝑊):𝑊 𝑟] ∪ [𝑊 − 𝑟𝑎𝑑(𝐾) + 𝑠𝑜𝑐(𝑊):𝑊 𝑠]. 

Proof: 

      (⟹) Let 𝑥 ∈ [𝐾:𝑊 𝑟𝑠], where 𝑟, 𝑠 ∈ 𝑅 and 𝑟𝑠 ∉ [𝐾 + 𝑠𝑜𝑐(𝑊):𝑅 𝑊], implies that 

𝑟𝑠𝑥 ∈ 𝐾. But 𝐾 is a pseudo primary-2-absorbing submodule of 𝑊, and 𝑟𝑠 ∉ [𝐾 +

𝑠𝑜𝑐(𝑊):𝑅 𝑊], then 𝑟𝑥 ∈ 𝑊 − 𝑟𝑎𝑑(𝐾) + 𝑠𝑜𝑐(𝑊) or 𝑠𝑥 ∈ 𝑊 − 𝑟𝑎𝑑(𝐾) + 𝑠𝑜𝑐(𝑊). That is 

either 𝑥 ∈ [𝑊 − 𝑟𝑎𝑑(𝐾) + 𝑠𝑜𝑐(𝑊):𝑊 𝑟] or 𝑥 ∈ [𝑊 − 𝑟𝑎𝑑(𝐾) + 𝑠𝑜𝑐(𝑊):𝑊 𝑠], thus 𝑥 ∈

[𝑊 − 𝑟𝑎𝑑(𝐾) + 𝑠𝑜𝑐(𝑊):𝑊 𝑟] ∪ [𝑊 − 𝑟𝑎𝑑(𝐾) + 𝑠𝑜𝑐(𝑊):𝑊 𝑠]. Hence [𝐾:𝑊 𝑟𝑠] ⊆

[𝑊 − 𝑟𝑎𝑑(𝐾):𝑊 𝑟] ∪ [𝑊 − 𝑟𝑎𝑑(𝐾):𝑊 𝑠]. 

       ( ⟸) Let  𝑟𝑠𝑥 ∈ 𝐾, where 𝑥 ∈ 𝑊 and 𝑟, 𝑠 ∈ 𝑅 with 𝑟𝑠 ∉ [𝐾 + 𝑠𝑜𝑐(𝑊):𝑅 𝑊]. It follows 

that 𝑥 ∈ [𝐾:𝑊 𝑟𝑠], by hypothesis 𝑥 ∈ [𝑊 − 𝑟𝑎𝑑(𝐾) + 𝑠𝑜𝑐(𝑊):𝑊 𝑟] ∪ [𝑊 − 𝑟𝑎𝑑(𝐾) +

𝑠𝑜𝑐(𝑊):𝑊 𝑠]. Hence 𝑥 ∈ [𝑊 − 𝑟𝑎𝑑(𝐾) + 𝑠𝑜𝑐(𝑊):𝑊 𝑟] or 𝑥 ∈ [𝑊 − 𝑟𝑎𝑑(𝐾) +

𝑠𝑜𝑐(𝑊):𝑊 𝑠]. Therefore 𝑟𝑥 ∈ 𝑊 − 𝑟𝑎𝑑(𝐾) + 𝑠𝑜𝑐(𝑊) or 𝑠𝑥 ∈ 𝑊 − 𝑟𝑎𝑑(𝐾) + 𝑠𝑜𝑐(𝑊), that 

is 𝐾 is a pseudo primary-2-absorbing submodule of 𝑊. 
 

Proposition (4) 

       Let 𝑊 be an 𝑅-module and 𝐿 be a proper submodule of 𝑊. Then 𝐿 is a pseudo primary-

2-absorbing submodule of 𝑊 if and only if 𝑟𝑠𝐾 ⊆ 𝐿 for 𝑟, 𝑠 ∈ 𝑅 and 𝐾 is a submodule of 𝑊, 

with 𝑟𝑠 ∉ [𝐿 + 𝑠𝑜𝑐(𝑊):𝑅 𝑊], implies that 𝑟𝐾 ⊆ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊) or 𝑠𝐾 ⊆ 𝑊 −

𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊). 

Proof 

      (⟹) Let 𝐿 be a pseudo primary-2-absorbing submodule of 𝑊, and 𝑟𝑠𝐾 ⊆ 𝐿, with  𝑟, 

𝑠 ∈ 𝑅 and 𝐾 is a submodule of 𝑊 with 𝑟𝑠 ∉ [𝐿 + 𝑠𝑜𝑐(𝑊):𝑅 𝑊]. Assume that 𝑟𝐾 ⊈ 𝑊 −

𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊) and 𝑠𝐾 ⊈ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊), then 𝑟𝑘1 ∉ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊) 

and 𝑠𝑘2 ∉ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊) for some 𝑘1, 𝑘2 ∈ 𝐾. Now we have 𝑟𝑠𝑘1 ∈ 𝐿 and since 𝐿 

is a pseudo primary-2-absorbing submodule of 𝑊 and 𝑟𝑠 ∉ [𝐿 + 𝑠𝑜𝑐(𝑊):𝑅 𝑊] and 𝑟𝑘1 ∉

𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊), then 𝑠𝑘1 ∈ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊). Also, since 𝑟𝑠𝑘2 ∈ 𝐿 and                



  

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𝑟𝑠 ∉ [𝐿 + 𝑠𝑜𝑐(𝑊):𝑅 𝑊] and 𝑠𝑘2 ∉ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊), then 𝑟𝑘2 ∈ 𝑊 − 𝑟𝑎𝑑(𝐿) +

𝑠𝑜𝑐(𝑊). Again since 𝑟𝑠(𝑘1 + 𝑘2) ∈ 𝐿 and 𝑟𝑠 ∉ [𝐿 + 𝑠𝑜𝑐(𝑊):𝑅 𝑊] we have 𝑟(𝑘1 + 𝑘2) ∈

 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊) or 𝑠(𝑘1 + 𝑘2) ∈  𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊). Suppose that 𝑟(𝑘1 +

𝑘2) = 𝑟𝑘1 + 𝑟𝑘2 ∈  𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊), but 𝑟𝑘2 ∈ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊), it follows 

that 𝑟𝑘1 ∈ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊) a contradiction. Suppose that 𝑠(𝑘1 + 𝑘2) = 𝑠𝑘1 + 𝑠𝑘2 ∈

 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊), but 𝑠𝑘1 ∈ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊), we have 𝑠𝑘2 ∈ 𝑊 − 𝑟𝑎𝑑(𝐿) +

𝑠𝑜𝑐(𝑊) a contradiction. Hence 𝑟𝐾 ⊆ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊) or 𝑠𝐾 ⊆ 𝑊 − 𝑟𝑎𝑑(𝐿) +

𝑠𝑜𝑐(𝑊). 

       ( ⟸) Let  𝑟𝑠𝑥 ∈ 𝐿, where 𝑥 ∈ 𝑊 and 𝑟, 𝑠 ∈ 𝑅 with 𝑟𝑠 ∉ [𝐿 + 𝑠𝑜𝑐(𝑊):𝑅 𝑊]. So 

𝑟𝑠(𝑥) ⊆ 𝐿, it follows by hypothesis 𝑟(𝑥) ⊆  𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊) or 𝑠(𝑥) ⊆  𝑊 −

𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊). That is 𝑟𝑥 ∈  𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊) or 𝑠𝑥 ∈  𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊). 

Hence 𝐿 is a pseudo primary-2-absorbing submodule of 𝑊. 
 

Proposition (5) 

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

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

submodule of 𝑊, implies that either 𝐼𝐽 ⊆ [𝐿 + 𝑠𝑜𝑐(𝑊):𝑅 𝑊] or 𝐼𝐾 ⊆ 𝑊 − 𝑟𝑎𝑑(𝐿) +

𝑠𝑜𝑐(𝑊) or 𝐽𝐾 ⊆ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊). 

Proof 

      (⟹)Assume  that 𝐿 is a pseudo primary-2-absorbing submodule of 𝑊, and 𝐼𝐽𝐾 ⊆ 𝐿, 

where 𝐼, 𝐽 are ideals of 𝑅 and 𝐾 is a submodule of 𝑊 and 𝐼𝐽 ⊈ [𝐿 + 𝑠𝑜𝑐(𝑊):𝑅 𝑊]. We must 

prove that 𝐼𝐾 ⊆ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊) or 𝐽𝐾 ⊆ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊). Suppose that 

𝐼𝐾 ⊈ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊) and 𝐽𝐾 ⊈ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊), it follows that there exists 

𝑟1 ∈ 𝐼 and 𝑟2 ∈ 𝐽 such that 𝑟1𝐾 ⊈ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊) and 𝑟2𝐾 ⊈ 𝑊 − 𝑟𝑎𝑑(𝐿) +

𝑠𝑜𝑐(𝑊). Now 𝑟1𝑟2𝐾 ⊆ 𝐿 with 𝑟1𝐾 ⊈ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊) and 𝑟2𝐾 ⊈ 𝑊 − 𝑟𝑎𝑑(𝐿) +

𝑠𝑜𝑐(𝑊) and 𝐿 is a pseudo primary-2-absorbing submodule of 𝑊, implies that by 

Proposition(4) 𝑟1𝑟2 ∈ [𝐿 + 𝑠𝑜𝑐(𝑊):𝑅 𝑊]. Since 𝐼𝐽 ⊈ [𝐿 + 𝑠𝑜𝑐(𝑊):𝑅 𝑊], it follows that there 

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

[𝐿 + 𝑠𝑜𝑐(𝑊):𝑅 𝑊], we have by Proposition (4) either 𝑠1𝐾 ⊆ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊) or 

𝑠2𝐾 ⊆ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊). 

Now we discussed the following cases: 

Case one: Suppose that 𝑠1𝐾 ⊆ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊) but 𝑠2𝐾 ⊈ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊). 

Since 𝑟1𝑠2𝐾 ⊆ 𝐿 and 𝐿 is a pseudo primary-2-absorbing submodule of 𝑊 with 𝑠2𝐾 ⊈ 𝑊 −

𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊) and 𝑟1𝐾 ⊈ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊), implies that 𝑟1𝑠2 ∈ [𝐿 +

𝑠𝑜𝑐(𝑊):𝑅 𝑊] by Proposition(4). Also since 𝑠1𝐾 ⊆ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊) but 𝑟1𝐾 ⊈ 𝑊 −

𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊), it follows that (𝑟1 + 𝑠1)𝐾 ⊈ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊). Since (𝑟1 +

𝑠1)𝑠2𝐾 ⊆ 𝐿 and 𝑠2𝐾 ⊈ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊)and (𝑟1 + 𝑠1)𝐾 ⊈ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊) 

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

𝑠1𝑠2 ∈ [𝐿 + 𝑠𝑜𝑐(𝑊):𝑅 𝑊] and 𝑟1𝑠2 ∈ [𝐿 + 𝑠𝑜𝑐(𝑊):𝑅 𝑊], implies that 𝑠1𝑠2 ∈ [𝐿 +

𝑠𝑜𝑐(𝑊):𝑅 𝑊] a contradiction. 

Case two: If 𝑠2𝐾 ⊆ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊) but 𝑠1𝐾 ⊈ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊) in similarly 

steps of Case one we get a contradiction. 

Case three: Assume that 𝑠1𝐾 ⊆ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊) but 𝑠2𝐾 ⊆ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊). 

Now since 𝑠2𝐾 ⊆ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊) and 𝑟2𝐾 ⊈ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊), it follows that 

(𝑟2 + 𝑠2)𝐾 ⊈ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊). We have 𝑟1(𝑟2 + 𝑠2)𝐾 ⊆ 𝐿 and 𝑟1𝐾 ⊈ 𝑊 − 𝑟𝑎𝑑(𝐿) +



  

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𝑠𝑜𝑐(𝑊) and (𝑟2 + 𝑠2)𝐾 ⊈ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊),  by Proposition(4) 𝑟1(𝑟2 + 𝑠2) = 𝑟1𝑟2 +

𝑟1𝑠2 ∈ [𝐿 + 𝑠𝑜𝑐(𝑊):𝑅 𝑊]. But 𝑟1𝑟2 ∈ [𝐿 + 𝑠𝑜𝑐(𝑊):𝑅 𝑊] and 𝑟1𝑟2 + 𝑟1𝑠2 ∈ [𝐿 +

𝑠𝑜𝑐(𝑊):𝑅 𝑊], it follows that  𝑟1𝑠2 ∈ [𝐿 + 𝑠𝑜𝑐(𝑊):𝑅 𝑊]. Now, since 𝑠1𝐾 ⊆ 𝑊 − 𝑟𝑎𝑑(𝐿) +

𝑠𝑜𝑐(𝑊) and 𝑟1𝐾 ⊈ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊), implies that (𝑟1 + 𝑠1)𝐾 ⊈ 𝑊 − 𝑟𝑎𝑑(𝐿) +

𝑠𝑜𝑐(𝑊) since (𝑟1 + 𝑠1)𝑟2𝐾 ⊆ 𝐿 and 𝑟2𝐾 ⊈ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊) and (𝑟1 + 𝑠1)𝐾 ⊈ 𝑊 −

𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊), it follows that (𝑟1 + 𝑠1)𝑟2 = 𝑟1𝑟2 + 𝑠1𝑟2 ∈ [𝐿 + 𝑠𝑜𝑐(𝑊):𝑅 𝑊] by 

Proposition(4). Now, since 𝑟1𝑟2 ∈ [𝐿 + 𝑠𝑜𝑐(𝑊):𝑅 𝑊] and 𝑟1𝑟2 + 𝑠1𝑟2 ∈ [𝐿 + 𝑠𝑜𝑐(𝑊):𝑅 𝑊], 

implies that 𝑠1𝑟2 ∈ [𝐿 + 𝑠𝑜𝑐(𝑊):𝑅 𝑊] . Also , since (𝑟1 + 𝑠1)(𝑟2 + 𝑠2)𝐾 ⊆ 𝐿 and (𝑟1 +

𝑠1)𝐾 ⊈ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊) and (𝑟2 + 𝑠2)𝐾 ⊈ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊), it follows that  

(𝑟1 + 𝑠1)(𝑟2 + 𝑠2) = 𝑟1𝑟2 + 𝑟1𝑠2 + 𝑠1𝑟2 + 𝑠1𝑠2 ∈ [𝐿 + 𝑠𝑜𝑐(𝑊):𝑅 𝑊] by Proposition(4). 

Again since 𝑟1𝑟2, 𝑟1𝑠2, 𝑠1𝑟2 ∈ [𝐿 + 𝑠𝑜𝑐(𝑊):𝑅 𝑊], we get that 𝑠1𝑠2 ∈ [𝐿 + 𝑠𝑜𝑐(𝑊):𝑅 𝑊] a 

contradiction. Thus we have either 𝐼𝐾 ⊆ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊) or 𝐽𝐾 ⊆ 𝑊 − 𝑟𝑎𝑑(𝐿) +

𝑠𝑜𝑐(𝑊). 

       ( ⟸) Obvious.   
 

Proposition (6) 

     Let 𝐿 be a proper submodule of an 𝑅-module 𝑊, with 𝑊 − 𝑟𝑎𝑑(𝐿) is a prime submodule 

of 𝑊. Then 𝐿 is a pseudo primary-2-absorbing submodule of 𝑊. 

Proof 

     Suppose that 𝑟𝑠𝑥 ∈ 𝐿, where 𝑟, 𝑠 ∈ 𝑅, 𝑥 ∈ 𝑊 and 𝑠𝑥 ∉ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊). Since 

𝐿 ⊆ 𝑊 − 𝑟𝑎𝑑(𝐿), then 𝑟(𝑠𝑥) ∈  𝑊 − 𝑟𝑎𝑑(𝐿), but 𝑊 − 𝑟𝑎𝑑(𝐿) is a prime submodule of 𝑊, 

then 𝑟𝑊 ⊆ 𝑊 − 𝑟𝑎𝑑(𝐿) ⊆ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊). That is 𝑟𝑥 ∈  𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊), 

for some 𝑥 ∈ 𝑊. Thus 𝐿 is a pseudo primary-2-absorbing submodule of 𝑊. 
 

Lemma (7)[11, Ex. 10, p. 29] 

        Let 𝐿 be an essential submodule of an 𝑅-module 𝑊, then 𝑠𝑜𝑐(𝐿) = 𝑠𝑜𝑐(𝑊). 
 

Proposition (8) 

       Let 𝐿 and 𝐾 are proper submodules of an 𝑅-module 𝑊 such that 𝐿 ⊊ 𝐾 and 𝐾 is an 

essential submodule of 𝑊. If 𝐿 is a pseudo primary-2-absorbing submodule of 𝑊, then  𝐿 is a 

pseudo primary-2-absorbing submodule of 𝐾. 

Proof 

       Suppose that 𝑟𝑠𝑥 ∈ 𝐿, where 𝑟, 𝑠 ∈ 𝑅, 𝑥 ∈ 𝐾 ⊆ 𝑊. Since 𝐿 is a pseudo primary-2-

absorbing submodule of 𝑊, implies that either 𝑟𝑥 ∈ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊) or 𝑠𝑥 ∈ 𝑊 −

𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊) or 𝑟𝑠𝑊 ⊆ 𝐿 + 𝑠𝑜𝑐(𝑊). But 𝐾 is an essential submodule of 𝑊, then by 

Lemma(7) 𝑠𝑜𝑐(𝐾) = 𝑠𝑜𝑐(𝑊). Hence we have either 𝑟𝑥 ∈ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝐾) or 

𝑠𝑥 ∈ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝐾) or 𝑟𝑠𝑊 ⊆ 𝐿 + 𝑠𝑜𝑐(𝐾). Thus 𝐿 is a pseudo primary-2-absorbing 

submodule of 𝐾. 

Before we introduce the next result we need to recall the following lemmas. 
 

Lemma (9)[13, Lemma(2.3.15)] 

      Let 𝐿, 𝐾 and 𝐷 are submodules of an 𝑅-module 𝑊 with 𝐾 ⊆ 𝐷, then (𝐿 + 𝐾) ∩ 𝐷 =

(𝐿 ∩ 𝐷) + 𝐾 = (𝐿 ∩ 𝐷) + (𝐾 ∩ 𝐷). 
 

Lemma (10)[14, Coro(9.9)] 

       Let 𝐾 be a submodule of an 𝑅-module 𝑊, then 𝑠𝑜𝑐(𝐾) = 𝐾 ∩ 𝑠𝑜𝑐(𝑊).    



  

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Proposition (11) 

       Let 𝐿 and 𝐾 be a proper submodules of an 𝑅-module 𝑊 with 𝐿 ⊊ 𝐾 and 𝑠𝑜𝑐(𝑊) ⊆ 𝐾. If 

𝐿 is a pseudo primary-2-absorbing submodule of 𝑊, then  𝐿 is a pseudo primary-2-absorbing 

submodule of 𝐾. 

Proof 

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

submodule of 𝑊, implies that either 𝑟𝑥 ∈ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊) or 𝑠𝑥 ∈ 𝑊 − 𝑟𝑎𝑑(𝐿) +

𝑠𝑜𝑐(𝑊) or 𝑟𝑠𝑊 ⊆ 𝐿 + 𝑠𝑜𝑐(𝑊). That is either 𝑟𝑥 ∈ (𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊)) ∩ 𝐾 or         

𝑠𝑥 ∈ (𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊)) ∩ 𝐾 or 𝑟𝑠𝑊 ⊆ (𝐿 + 𝑠𝑜𝑐(𝑊)) ∩ 𝐾. But by Lemma (9) 

(𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊)) ∩ 𝐾 = (𝑊 − 𝑟𝑎𝑑(𝐿) ∩ 𝐾) + (𝑠𝑜𝑐(𝑊) ∩ 𝐾) = (𝑊 − 𝑟𝑎𝑑(𝐿) ∩

𝐾) + 𝑠𝑜𝑐(𝐾) by Lemma(10). Thus we have either 𝑟𝑥 ∈ ( 𝑊 − 𝑟𝑎𝑑(𝐿) ∩ 𝐾) + 𝑠𝑜𝑐(𝐾) ⊆

𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝐾) or 𝑠𝑥 ∈ ( 𝑊 − 𝑟𝑎𝑑(𝐿) ∩ 𝐾) + 𝑠𝑜𝑐(𝐾) ⊆ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝐾) or 

𝑟𝑠𝑊 ⊆ (𝐿 ∩ 𝐾) + (𝑠𝑜𝑐(𝑊) ∩ 𝐾) = (𝐿 ∩ 𝐾) + 𝑠𝑜𝑐(𝐾) ⊆ 𝐿 + 𝑠𝑜𝑐(𝐾). Hence 𝐿 is a pseudo 

primary-2-absorbing submodule of 𝐾. 

Recall that for any submodules 𝐿, 𝐾 of a multiplication 𝑅-module 𝑊 with 𝐿 = 𝐼𝑊, 𝐾 = 𝐽𝑊 

for some ideals 𝐼 and 𝐽 of 𝑅. The product 𝐿𝐾 = 𝐼𝑊. 𝐽𝑊 = 𝐼𝐽𝑊. That is 𝐿𝐾 = 𝐼𝐾, in 

particular 𝐿𝑊 = 𝐼𝑊𝑊 = 𝐼𝑊 = 𝐿. Also for any 𝑥 ∈ 𝑊 we have 𝐿𝑥 = 𝐼𝑥[15]. 

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

class of multiplication modules. 
 

Proposition (12) 

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

pseudo primary-2-absorbing submodule of 𝑊 if and only if, whenever  𝐿1𝐿2𝐿3 ⊆ 𝐿 for 𝐿1, 𝐿2, 

𝐿3 are submodules of 𝑊, implies that either 𝐿1𝐿3 ⊆ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊) or 𝐿2𝐿3 ⊆ 𝑊 −

𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊) or 𝐿1𝐿2𝑊 ⊆ 𝐿 + 𝑠𝑜𝑐(𝑊). 

Proof 

       (⟹) Let 𝐿 be is a pseudo primary-2-absorbing submodule of 𝑊 and 𝐿1𝐿2𝐿3 ⊆ 𝐿 for 𝐿1, 

𝐿2, 𝐿3 are submodules of 𝑊, with 𝐿1𝐿2𝑊 ⊈ 𝐿 + 𝑠𝑜𝑐(𝑊). Since 𝑊 is a multiplication, then 

𝐿1 = 𝐼1𝑊 and 𝐿2 = 𝐼2𝑊 for some ideals 𝐼1, 𝐼2, 𝐼3 of 𝑅. Clearly 𝐼1𝐼2𝐿3 ⊆ 𝐿 and 𝐼1𝐼2 ⊈

[𝐿 + 𝑠𝑜𝑐(𝑊):𝑅 𝑊]. Since 𝐿 is a pseudo primary-2-absorbing submodule of 𝑊, implies that 

either 𝐼1𝐿3 ⊆ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊) or 𝐼2𝐿3 ⊆ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊), it follows that 

either 𝐿1𝐿3 ⊆ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊)  or 𝐿2𝐿3 ⊆ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊). 

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

Since 𝑊 is multiplication, then 𝐼1𝐼2𝐾 = 𝐿1𝐿2𝐾 ⊆ 𝐿, by hypothesis either 𝐿1𝐾 ⊆  𝑊 −

𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊) or 𝐿2𝐾 ⊆  𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊) or 𝐿1𝐿2 ⊆ [𝐿 + 𝑠𝑜𝑐(𝑊):𝑅 𝑊]. That is 

either 𝐼1𝐾 ⊆  𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊) or 𝐼2𝐾 ⊆  𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊) or 𝐼1𝐼2 ⊆

[𝐿 + 𝑠𝑜𝑐(𝑊):𝑅 𝑊]. Then by Proposition (5) 𝐿 is a pseudo primary-2-absorbing submodule of 

𝑊. 
 

Lemma (13)[2]. 

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

𝑓(𝑊 − 𝑟𝑎𝑑(𝐿)) = �̅� − 𝑟𝑎𝑑(𝑓(𝐿)). 
 

 

 



  

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Proposition (14) 

     Let 𝑓: 𝑊 ⟶ �̅� be an 𝑅-epimorphism and �̅� is a pseudo primary-2-absorbing submodule 

of �̅�. Then 𝑓 −1(�̅�) is a pseudo primary-2-absorbing submodule of 𝑊. 

Proof 

    Let ∈ 𝑓 −1(�̅�) , where 𝑟, 𝑠 ∈ 𝑅, 𝑥 ∈ 𝑊, implies that 𝑟𝑠𝑓(𝑥) ∈  �̅�. Since �̅� is a pseudo 

primary-2-absorbing submodule of �̅�, it follows that either 𝑟𝑓(𝑥) ∈  �̅� − 𝑟𝑎𝑑(�̅�) + 𝑠𝑜𝑐(�̅�) 

or 𝑠𝑓(𝑥) ∈  �̅� − 𝑟𝑎𝑑(�̅�) + 𝑠𝑜𝑐(�̅�) or 𝑠𝑟�̅� ⊆ �̅� + 𝑠𝑜𝑐(�̅�).Thus either 𝑟𝑥 ∈ 𝑓 −1(�̅� −

𝑟𝑎𝑑(�̅�)) + 𝑓 −1(𝑠𝑜𝑐(�̅�)) ⊆ 𝑊 − 𝑟𝑎𝑑(𝑓 −1(�̅�)) + 𝑠𝑜𝑐(𝑊) or 𝑠𝑥 ∈ 𝑓 −1(�̅� − 𝑟𝑎𝑑(�̅�)) +

𝑓 −1(𝑠𝑜𝑐(�̅�)) ⊆ 𝑊 − 𝑟𝑎𝑑(𝑓 −1(�̅�)) + 𝑠𝑜𝑐(𝑊) or 𝑟𝑠𝑊 ⊆ 𝑓 −1(�̅�) + 𝑠𝑜𝑐(𝑊). Hence 𝑓 −1(�̅�) 

be a pseudo primary-2-absorbing submodule of 𝑊. 
 

Proposition (15) 

     Let 𝑓: 𝑊 ⟶ �̅� be an 𝑅-epimorphism and 𝐿 is a pseudo primary-2-absorbing submodule 

of 𝑊 with ker (𝑓) ⊆ 𝐿 . Then 𝑓(𝐿) is a pseudo primary-2-absorbing submodule of �̅�. 

 Proof 

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

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

0, implies that 𝑟𝑠𝑥 − 𝑙 ∈ ker (𝑓) ⊆ 𝐿, then 𝑟𝑠𝑥 ∈ 𝐿. But  𝐿 be a pseudo primary-2-absorbing 

submodule of 𝑊, then either 𝑟𝑥 ∈ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊) or  𝑠𝑥 ∈ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊) 

or 𝑟𝑠𝑊 ⊆ 𝐿 + 𝑠𝑜𝑐(𝑊), it follows that by Lemma(13) either 𝑟𝑓(𝑥) ∈ 𝑓(𝑊 − 𝑟𝑎𝑑(𝐿)) +

𝑓(𝑠𝑜𝑐(𝑊)) ⊆ �̅� − 𝑟𝑎𝑑(𝑓(𝐿)) + 𝑠𝑜𝑐(�̅�) or 𝑠𝑓(𝑥) ∈ 𝑓(𝑊 − 𝑟𝑎𝑑(𝐿)) + 𝑓(𝑠𝑜𝑐(𝑊)) ⊆

�̅� − 𝑟𝑎𝑑(𝑓(𝐿)) + 𝑠𝑜𝑐(�̅�) or 𝑟𝑠𝑓(𝑊) ⊆ 𝑓(𝐿) + 𝑓(𝑠𝑜𝑐(𝑊)) ⊆ 𝑓(𝐿) + 𝑠𝑜𝑐(�̅�). That is 

either 𝑟�̅� ∈ �̅� − 𝑟𝑎𝑑(𝑓(𝐿)) + 𝑠𝑜𝑐(�̅�) or 𝑠�̅� ∈ �̅� − 𝑟𝑎𝑑(𝑓(𝐿)) + 𝑠𝑜𝑐(�̅�) or 𝑟𝑠�̅� ⊆

𝑓(𝐿) + 𝑠𝑜𝑐(�̅�). Hence 𝑓(𝐿) is a pseudo primary-2-absorbing submodule of �̅�. 
 

Lemma (16)[12, Theo(2.12)]. 

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

𝑅-module 𝑊 and 𝐴 = [𝐿:𝑅 𝑊]. Then𝑊 − 𝑟𝑎𝑑(𝐿) = √𝐴. 𝑊 = √[𝐿:𝑅 𝑊]. 𝑊. 
 

Lemma (17)[12, Coro(2.14)]. 

       Let 𝑊 be faithful multiplication 𝑅-module, then 𝑠𝑜𝑐(𝑅)𝑊 = 𝑠𝑜𝑐(𝑊). 
 

Proposition (18) 

     Let 𝑊 be a faithful multiplication 𝑅-module and 𝐿 is a proper submodule of 𝑊. Then 𝐿 is 

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

absorbing ideal of 𝑅. 

Proof 

     (⟹) Let 𝐿 is a pseudo primary-2-absorbing submodule of 𝑊, and 𝑟. 𝑠. 𝑡 ∈ [𝐿:𝑅 𝑊] for 

𝑟,𝑠,𝑡 ∈ 𝑅, implies that 𝑟𝑠𝑡𝑊 ⊆ 𝐿, that is 𝑟𝑠𝑡(𝑥) ∈ 𝐿 for all 𝑥 ∈ 𝑊. But 𝑊 is a multiplication 

𝑅-module, then (𝑥) = 𝐼𝑊  for some ideal 𝐼 of 𝑅. That is 𝑟𝑠(𝑡𝐼𝑊) ⊆ 𝐿. Since 𝐿 is a pseudo 

primary-2-absorbing submodule of 𝑊, then by Proposition (4) either 𝑟(𝑡𝐼𝑊) ⊆ 𝑊 −

𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊) or 𝑠(𝑡𝐼𝑊) ⊆ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊) or 𝑟𝑠𝑊 ⊆ 𝐿 + 𝑠𝑜𝑐(𝑊) by Lemma 

(16)  𝑊 − 𝑟𝑎𝑑(𝐿) = √[𝐿:𝑅 𝑊]. 𝑊 and by Lemma (17) 𝑠𝑜𝑐(𝑅)𝑊 = 𝑠𝑜𝑐(𝑊). Hence we get       

either 𝑟(𝑡𝐼𝑊) ⊆ √[𝐿:𝑅 𝑊]. 𝑊 + 𝑠𝑜𝑐(𝑅)𝑊 or 𝑠(𝑡𝐼𝑊) ⊆ √[𝐿:𝑅 𝑊]. 𝑊 + 𝑠𝑜𝑐(𝑅)𝑊 or               



  

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𝑟𝑠𝑊 ⊆ [𝐿:𝑅 𝑊]𝑊 + 𝑠𝑜𝑐(𝑅)𝑊. That is either 𝑟𝑡𝑥 ∈ √[𝐿:𝑅 𝑊]. 𝑊 + 𝑠𝑜𝑐(𝑅)𝑊 or 𝑠𝑡𝑥 ∈

√[𝐿:𝑅 𝑊]. 𝑊 + 𝑠𝑜𝑐(𝑅)𝑊 for all 𝑥 ∈ 𝑊 or  𝑟𝑠𝑊 ⊆ [𝐿:𝑅 𝑊]𝑊 + 𝑠𝑜𝑐(𝑅)𝑊. It follows that 

either 𝑟𝑡𝑊 ⊆ √[𝐿:𝑅 𝑊]. 𝑊 + 𝑠𝑜𝑐(𝑅)𝑊 or 𝑠𝑡𝑊 ⊆ √[𝐿:𝑅 𝑊]. 𝑊 + 𝑠𝑜𝑐(𝑅)𝑊 or 𝑟𝑠𝑊 ⊆

[𝐿:𝑅 𝑊]𝑊 + 𝑠𝑜𝑐(𝑅)𝑊. Hence either 𝑟𝑡 ∈ √[𝐿:𝑅 𝑊] + 𝑠𝑜𝑐(𝑅) or 𝑠𝑡 ∈ √[𝐿:𝑅 𝑊] + 𝑠𝑜𝑐(𝑅) 

or 𝑟𝑠 ∈ [𝐿:𝑅 𝑊] + 𝑠𝑜𝑐(𝑅). Therefore [𝐿:𝑅 𝑊] is pseudo primary-2-absorbing ideal of 𝑅. 

       (⟸) Assume that [𝐿:𝑅 𝑊] is pseudo primary-2-absorbing ideal of 𝑅, and 𝑟𝑠𝑥 ∈ 𝐿, for 

𝑟,𝑠 ∈ 𝑅, 𝑥 ∈ 𝑊, that is 𝑟𝑠(𝑥) ⊆ 𝐿. Since 𝑊 is a multiplication 𝑅-module then (𝑥) = 𝐼𝑊 for 

some ideal 𝐼 of 𝑅. Thus 𝑟𝑠𝐼𝑊 ⊆ 𝐿, implies that 𝑟𝑠𝐼 ⊆ [𝐿:𝑅 𝑊]. By hypothesis and 

Proposition (4) either 𝑟𝐼 ⊆ √[𝐿:𝑅 𝑊] + 𝑠𝑜𝑐(𝑅) or 𝑠𝐼 ⊆ √[𝐿:𝑅 𝑊] + 𝑠𝑜𝑐(𝑅) or 𝑟𝑠 ∈

[𝐿:𝑅 𝑊] + 𝑠𝑜𝑐(𝑅). That is either 𝑟𝐼𝑊 ⊆ √[𝐿:𝑅 𝑊]𝑊 + 𝑠𝑜𝑐(𝑅)𝑊 or 𝑠𝐼𝑊 ⊆ √[𝐿:𝑅 𝑊]𝑊 +

𝑠𝑜𝑐(𝑅)𝑊  or 𝑟𝑠𝑊 ⊆ [𝐿:𝑅 𝑊]𝑊 + 𝑠𝑜𝑐(𝑅)𝑊. Thus by Lemma (16) and Lemma (17) we get   

𝑟𝑥 ∈ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊)  or 𝑠𝑥 ∈ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊)  or 𝑟𝑠𝑊 ⊆ 𝐿 + 𝑠𝑜𝑐(𝑊). 

Hence 𝐿 is a pseudo primary-2-absorbing submodule of 𝑊. 

 We need to recall the following lemma before we introduce the next result. 
 

Lemma (19)[11, Coro(1.26)]. 

       If 𝑊 is a non-singular 𝑅-modules, then 𝑠𝑜𝑐(𝑅)𝑊 = 𝑠𝑜𝑐(𝑊). 
 

 Proposition (20) 

       Let 𝑊 be a non-singular multiplication 𝑅-module and 𝐿 is a proper submodule of 𝑊. 

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

primary-2-absorbing ideal of 𝑅. 

Proof 

      (⟸) Let [𝐿:𝑅 𝑊] is a pseudo primary-2-absorbing ideal of 𝑅, and 𝑎𝑏𝑦 ∈ 𝐿, for 𝑎,𝑏 ∈ 𝑅, 

𝑦 ∈ 𝑊, that is 𝑎𝑏(𝑦) ⊆ 𝐿, it follows that 𝑎𝑏𝐽𝑊 ⊆ 𝐿 for 𝑊 is a multiplication 𝑅-module. 

Hence 𝑎𝑏𝐽 ⊆ [𝐿:𝑅 𝑊], implies that by Proposition (4) either 𝑎𝐽 ⊆ √[𝐿:𝑅 𝑊] + 𝑠𝑜𝑐(𝑅) or 

𝑏𝐽 ⊆ √[𝐿:𝑅 𝑊] + 𝑠𝑜𝑐(𝑅) or 𝑎𝑏 ∈ [[𝐿:𝑅 𝑊] + 𝑠𝑜𝑐(𝑅): 𝑅] = [𝐿:𝑅 𝑊] + 𝑠𝑜𝑐(𝑅). Thus either 

𝑎𝐽𝑊 ⊆ √[𝐿:𝑅 𝑊]𝑊 + 𝑠𝑜𝑐(𝑅)𝑊 or 𝑏𝐽𝑊 ⊆ √[𝐿:𝑅 𝑊]𝑊 + 𝑠𝑜𝑐(𝑅)𝑊 or 𝑎𝑏𝑊 ⊆

[𝐿:𝑅 𝑊]𝑊 + 𝑠𝑜𝑐(𝑅)𝑊. Hence by Lemma (16) and Lemma (19), we have either 𝑎𝑦 ∈ 𝑊 −

𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊) or 𝑏𝑦 ∈ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊) or 𝑎𝑏𝑊 ⊆ 𝐿 + 𝑠𝑜𝑐(𝑊). Therefore 𝐿 is a 

pseudo primary-2-absorbing submodule of 𝑊. 

       (⟹) Let it be 𝑎𝑏𝑐 ∈ [𝐿:𝑅 𝑊] where 𝑎,𝑏,𝑐 ∈ 𝑅, then 𝑎𝑏𝑐𝑊 ⊆ 𝐿, so 𝑎𝑏𝑐𝑦 ∈ 𝐿 for all 

𝑦 ∈ 𝑊. Since 𝑊 is a multiplication𝑅-module, then (𝑦) = 𝐽𝑊, thus 𝑎𝑏𝑐(𝑦) ⊆ 𝐿, it follows 

that 𝑎𝑏(𝑐𝐽𝑊) ⊆ 𝐿, implies that by hypothesis and by Proposition(4) either 𝑎(𝑐𝐽𝑊) ⊆ 𝑊 −

𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊) or 𝑏(𝑐𝐽𝑊) ⊆ 𝑊 − 𝑟𝑎𝑑(𝐿) + 𝑠𝑜𝑐(𝑊) or 𝑎𝑏𝑊 ⊆ 𝐿 + 𝑠𝑜𝑐(𝑊). It follows 

that by Lemma (16) and by Lemma (19) and 𝑊 is multiplication either 𝑎𝑐𝑦 ∈ √[𝐿:𝑅 𝑊]𝑊 +

𝑠𝑜𝑐(𝑅)𝑊 or 𝑏𝑐𝑦 ∈ √[𝐿:𝑅 𝑊]𝑊 + 𝑠𝑜𝑐(𝑅)𝑊 for all 𝑦 ∈ 𝑊 or 𝑎𝑏𝑊 ⊆ [𝐿:𝑅 𝑊]𝑊 +

𝑠𝑜𝑐(𝑅)𝑊. Hence either 𝑎𝑐𝑊 ⊆ √[𝐿:𝑅 𝑊]𝑊 + 𝑠𝑜𝑐(𝑅)𝑊 or 𝑏𝑐𝑊 ⊆ √[𝐿:𝑅 𝑊]𝑊 +

𝑠𝑜𝑐(𝑅)𝑊 or 𝑎𝑏𝑊 ⊆ [𝐿:𝑅 𝑊]𝑊 + 𝑠𝑜𝑐(𝑅)𝑊. That is either 𝑎𝑐 ∈ √[𝐿:𝑅 𝑊] + 𝑠𝑜𝑐(𝑅) 

or 𝑏𝑐 ∈ √[𝐿:𝑅 𝑊] + 𝑠𝑜𝑐(𝑅) or 𝑎𝑏 ∈ [𝐿:𝑅 𝑊] + 𝑠𝑜𝑐(𝑅) = [[𝐿:𝑅 𝑊] + 𝑠𝑜𝑐(𝑅): 𝑅]. That is 

[𝐿:𝑅 𝑊] is a pseudo primary-2-absorbing ideal of 𝑅. 

We need to recall the following results before we introduce the next propositions. 



  

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Lemma (21)[16, Coro of Theo. 9] 

         Let 𝐼1 and 𝐼2 are ideals of a ring 𝑅 and 𝑊 is a finitely generated multiplication 𝑅-

module. Then 𝐼1𝑊 ⊆ 𝐼2𝑊 if and only if 𝐼1 ⊆ 𝐼2 + 𝑎𝑛𝑛𝑅 (𝑊). 
 

Lemma (22)[17, Pro. (2.4)]. 

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

𝑊 − 𝑟𝑎𝑑(𝐼𝑊) = √𝐼𝑊. 
 

Proposition (23) 

     Let 𝑊 be a faithful finitely generated multiplication 𝑅-module and 𝐼 is a pseudo primary-

2-absorbing ideal of 𝑅 and 𝐼𝑊 ≠ 𝑊. Then 𝐼𝑊 is a pseudo primary-2-absorbing submodule of 

𝑊. 

Proof 

      Let 𝑎𝑏𝑥 ∈ 𝐼𝑊 for 𝑎,𝑏 ∈ 𝑅, 𝑥 ∈ 𝑊, then 𝑎𝑏(𝑥) ⊆ 𝐼𝑊, implies that 𝑎𝑏𝐽𝑊 ⊆ 𝐼𝑊 for some 

ideal 𝐽 of 𝑅 since 𝑊 is a multiplication. Hence by Lemma(21) 𝑎𝑏𝐽 ⊆ 𝐼 + 𝑎𝑛𝑛𝑅 (𝑊), but 𝑊 is 

a faithful. It follows that 𝑎𝑛𝑛𝑅 (𝑊) = (0), that is 𝑎𝑏𝐽 ⊆ 𝐼. Since 𝐼 is a pseudo primary-2-

absorbing ideal of 𝑅, then by Proposition (4) either 𝑎𝐽 ⊆ √𝐼 + 𝑠𝑜𝑐(𝑅) or 𝑏𝐽 ⊆ √𝐼 + 𝑠𝑜𝑐(𝑅) 

or 𝑎𝑏 ∈ [𝐼 + 𝑠𝑜𝑐(𝑅): 𝑅] = 𝐼 + 𝑠𝑜𝑐(𝑅). It follows that 𝑎𝐽𝑊 ⊆ √𝐼𝑊 + 𝑠𝑜𝑐(𝑅)𝑊 or 𝑏𝐽𝑊 ⊆

√𝐼𝑊 + 𝑠𝑜𝑐(𝑅)𝑊  or 𝑎𝑏𝑊 ⊆ 𝐼𝑊 + 𝑠𝑜𝑐(𝑅)𝑊. But by Lemma (17)𝑠𝑜𝑐(𝑅)𝑊 = 𝑠𝑜𝑐(𝑊) and 

by Lemma (22)  √𝐼𝑊 =. 𝑊 − 𝑟𝑎𝑑(𝐼𝑊). Thus either 𝑎𝑥 ∈ 𝑊 − 𝑟𝑎𝑑(𝐼𝑊) + 𝑠𝑜𝑐(𝑊) or 

𝑏𝑥 ∈ 𝑊 − 𝑟𝑎𝑑(𝐼𝑊) + 𝑠𝑜𝑐(𝑊) or 𝑎𝑏𝑊 ⊆ 𝐼𝑊 + 𝑠𝑜𝑐(𝑊). Hence 𝐼𝑊 is a pseudo primary-2-

absorbing submodule of 𝑊. 
 

Proposition (24) 

       Let 𝑊 be a faithful finitely generated multiplication 𝑅-module and 𝐾 be a proper 

submodule of 𝑊. Then the following statements are equivalent . 

1. K is a pseudo primary-2-absorbing submodule of 𝑊. 

2. [K:𝑅 W] is a pseudo primary-2-absorbing ideal of 𝑅. 

3. K = JW for some pseudo primary-2-absorbing ideal of 𝑅. 

Proof 

(1) ⇔ (2) By Proposition (18). 

(2) ⇒ (3) Since [K:𝑅 W] is a pseudo primary-2-absorbing ideal of 𝑅 with 𝑎𝑛𝑛𝑅 (𝑊) =

[0: W] ⊆ [K:𝑅 W] and K = [K:𝑅 W]W, implies that K = IW where I = [K:𝑅 W] is a pseudo 

primary-2-absorbing ideal of 𝑅. 

(3) ⟹ (2) Suppose that 𝐾 = 𝐽𝑊 for some a pseudo primary-2-absorbing ideal 𝐽 of 𝑅. Since 

𝑊 is multiplication, then K = [K:𝑅 W]W = IW. Since 𝑊 is faithful finitely generated 

multiplication then we have [K:𝑅 W] = J. Thus [K:𝑅 W] is a pseudo primary-2-absorbing ideal 

of 𝑅. 
 

Proposition (25) 

       Let 𝑊 be a finitely generated multiplication non-singular 𝑅-module and 𝐼 be a pseudo 

primary-2-absorbing ideal of 𝑅 with 𝑎𝑛𝑛𝑅 (𝑊) ⊆ 𝐼. Then 𝐼𝑊 is a pseudo primary-2-

absorbing submodule of 𝑊.  

Proof 

      Similarly as in Proposition (23) and using Lemma (19). 

 



  

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Proposition (26) 

       Let 𝑊 be a finitely generated multiplication non-singular 𝑅-module and 𝐾 be a proper 

submodule of 𝑊. Then the following statements are equivalent. 

1. K is a pseudo primary-2-absorbing submodule of 𝑊. 

2. [K:𝑅 W] is a pseudo primary-2-absorbing ideal of 𝑅. 

3. K = JW for some pseudo primary-2-absorbing ideal of 𝑅, with 𝑎𝑛𝑛𝑅 (𝑊) ⊆ 𝐽. 

Proof 

      Similarly as in Proposition (24), by using Proposition (20). 
 

3. Conclusion 

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

2-absorbing) submodules called pseudo primary-2-absorbing submodules and we explain the 

converse implication of 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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