IHJPAS. 36(2)2023 

341 
This work is licensed under a Creative Commons Attribution 4.0 International License 

 

   

 

 

 

 

 

 

 

 

 
 

 

 

Abstract 

  

The concept of a 2-Absorbing submodule is considered as an essential feature in the field of 

module theory and has many generalizations. This articale discusses the concept of the Extend 

Nearly Pseudo Quasi-2-Absorbing submodules and their relationship to the 2-Absorbing 

submodule, Quasi-2-Absorbing submodule, Nearly-2-Absorbing submodule, Pseudo-2-Absorbing 

submodule, and the rest of the other concepts previously studied. The relationship between them 

has been studied, explaining that the opposite is not true and that under certain conditions the 

opposite becomes true. This article aims to study this concept and gives the most important 

propositions, characterizations, remarks, examples, lemmas, and observations related to it. In the 

end, we will present a very important equivalent of our concept with the rest of the concepts 

presented previously. 

  

Keywords: 2-Absorbing submodule, Quasi-2-Absorbing submodule,  socal of module, Jacobson 

of module, cyclic and multiplication modules. 

 

1. Introduction 

As a generalization of the 2-Absorbing Ideal, the 2-Absorbing Submodule notion was originally 

introduced in 2011 by Darani A. and Sohelinia F. Badawi A. first proposed the 2-Absorbing Ideal 

in 2007. In recent years, various generalizations of the 2-Absorbing submodule, including the 

Quasi-2-Absorbing submodule, have been introduced. In this article, we presented the Extend 

Nearly Pseudo Quasi-2-Absorbing submodule, a new generalization on previously studied 

concepts, particularly the 2-Absorbing submodule and Quasi-2-Absorbing submodule. It is worth 

noting that Ʀ is a commutative ring with a 𝑛onzero 𝑖dentity and  Ѡ be a unitary Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒.  In the 
end, we presented the most important propositions in this research and studied all possible 

relationships with this concept within the best conditions that helped us reach the best solutions. 

 

2.  Preliminaries 
In the following, we mention some basic definitions and notations in module that will be used in 

this paper. 

 

doi.org/10.30526/36.2.3019 

Article history: Received 16 September 2022, Accepted 6 November 2022, Published in April 2023. 

 

 

 

Ibn Al-Haitham Journal for Pure and Applied Sciences 

Journal homepage: jih.uobaghdad.edu.iq 

 

Extend Nearly Pseudo Quasi-2-Absorbing submodules(I) 

 

Omar A. Abdullah 
Department of Mathematics College of 

Computer Science and Mathematics 

Tikrit University / Iraq. 

omer.a.abdullah35383@st.tu.edu.iq 

 
 

 

 

 

Haibat K. Mohammadali 
Department of Mathematics College of 

Computer Science and Mathematics 

Tikrit University / Iraq. 
H.mohammadali@tu.edu.iq 

 

 

 

 
 

https://creativecommons.org/licenses/by/4.0/
mailto:omer.a.abdullah35383@st.tu.edu.iq
mailto:H.mohammadali@tu.edu.iq
mailto:H.mohammadali@tu.edu.iq


IHJPAS. 36(2)2023 

342 
 

Definition 2.1[1]. 

A 𝑝𝑟𝑜𝑝𝑒𝑟 submodule 𝑉 of 𝑎𝑛 Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒 Ѡ is called 2-Absorbing submodule if whenever ɑɓ𝑥 ∈
𝑉 for ɑ, ɓ ∈ Ʀ, 𝑥 ∈ Ѡ 𝑖𝑚𝑝𝑙𝑖𝑒𝑠 that either ɑ𝑥 ∈ 𝑉 or  ɓ𝑥 ∈ 𝑉 or  ɑɓ ∈ [𝑉:Ʀ Ѡ]. Where [𝑉:Ʀ Ѡ] =
{ɑ ∈ Ʀ: ɑѠ ⊆ 𝑉}[2].  

Definition 2.2[3]. 

𝑠𝑜𝑐(Ѡ) is the 𝑖𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡𝑖𝑜𝑛 of all 𝑒𝑠𝑠𝑒𝑛𝑡𝑖𝑎𝑙 submodules of Ѡand a 𝑛𝑜𝑛𝑧𝑒𝑟𝑜 submodule 𝑉 of Ѡ 
is 𝑒𝑠𝑠𝑒𝑛𝑡𝑖𝑎𝑙 in Ѡ if 𝑉 ∩ 𝐺 ≠ (0) for any 𝑛𝑜𝑛𝑧𝑒𝑟𝑜 submodule 𝐺 of Ѡ. 

Definition 2.3[4]. 

A submodule 𝑉 of 𝑎𝑛 Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒 Ѡ is called a 𝑚𝑎𝑥𝑖𝑚𝑎𝑙 submodule of Ѡ, if 𝑉 is a 𝑝𝑟𝑜𝑝𝑒𝑟 
submodule of Ѡ and for all 𝐵 ⊆ Ѡ with 𝑉 ⊂ 𝐵, then 𝐵 = Ѡ, 𝐽(Ѡ) is the intersection of all 
𝑚𝑎𝑥𝑖𝑚𝑎𝑙 submodules of Ѡ. 

Definition 2.4[5]. 

A 𝑝𝑟𝑜𝑝𝑒𝑟 submodule 𝑉 of 𝑎𝑛 Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒 Ѡ is called Nearly-2-Absorbing submodule if whenever 
𝑟𝑠𝑚 ∈  𝑉, for 𝑟, 𝑠 ∈  Ʀ, 𝑚 ∈  Ѡ, implies that either 𝑟𝑚 ∈  𝑉 +  𝐽(Ѡ) or 𝑠𝑚 ∈  𝑉 +  𝐽(Ѡ) or 
𝑟𝑠Ѡ ⊆  𝑉 +  𝐽(Ѡ). 

Definition 2.5[6]. 

A 𝑝𝑟𝑜𝑝𝑒𝑟 submodule 𝑉 of 𝑎𝑛 Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒 Ѡ is called Quasi-2-Absorbing submodule if whenever 
𝑟𝑠𝑡𝑚 ∈ 𝑉 for 𝑟, 𝑠, 𝑡 ∈  Ʀ, 𝑚 ∈  Ѡ, implies that either 𝑟𝑡𝑚 ∈  𝑉 or 𝑠𝑡𝑚 ∈  𝑉 or 𝑟𝑠Ѡ ⊆  𝑉. 

Definition 2.6[7]. 

A 𝑝𝑟𝑜𝑝𝑒𝑟 submodule 𝑉 of an Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒 Ѡ is called Pseudo-2-Absorbing submodule if 
whenever 𝑟𝑠𝑚 ∈ 𝑉 for 𝑟, 𝑠 ∈  Ʀ, 𝑚 ∈  Ѡ, 𝑖𝑚𝑝𝑙𝑖𝑒𝑠 that either 𝑟𝑚 ∈  𝑉 +  𝑠𝑜𝑐(Ѡ) or 𝑠𝑚 ∈
 𝑉 +  𝑠𝑜𝑐(Ѡ) or 𝑟𝑠Ѡ ⊆  𝑉 +  𝑠𝑜𝑐(Ѡ). And in the same paper [7] the concept of Pseudo Quasi-
2-Absorbing submodule is introduced, where a 𝑝𝑟𝑜𝑝𝑒𝑟 submodule 𝑉 of an Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒 Ѡ is called 
Pseudo Quasi-2-Absorbing submodule if for any 𝑟𝑠𝑡𝑚 ∈ 𝑉 for 𝑟, 𝑠, 𝑡 ∈  𝑅, 𝑚 ∈  Ѡ, 𝑖𝑚𝑝𝑙𝑖𝑒𝑠 that 
either 𝑟𝑠𝑚 ∈  𝑉 +  𝑠𝑜𝑐(Ѡ) or 𝑠𝑡𝑚 ∈  𝑉 +  𝑠𝑜𝑐(Ѡ) or 𝑟𝑡𝑚 ∈  𝑉 +  𝑠𝑜𝑐(Ѡ). 

Definition 2.7[8]. 

A 𝑝𝑟𝑜𝑝𝑒𝑟 submodule 𝑉 of an Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒 Ѡ is called Nearly Quasi-2-𝐴bsorbing submodule if 
whenever 𝑟𝑠𝑡𝑚 ∈  𝑉 for 𝑟, 𝑠, 𝑡 ∈  Ʀ, 𝑚 ∈  Ѡ, 𝑖𝑚𝑝𝑙𝑖𝑒𝑠 that either 𝑟𝑡𝑚 ∈  𝑉 +  𝐽(Ѡ) or 𝑠𝑡𝑚 ∈
 𝑉 +  𝐽(Ѡ) or 𝑟𝑠Ѡ ⊆  𝑉 +  𝐽(Ѡ).  

Definition 2.8[9]. 

An Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒 Ѡ is a 𝑠𝑒𝑚𝑖𝑠𝑖𝑚𝑝𝑙𝑒, if every submodule of Ѡ is a direct summand.  

Definition 2.9[4]. 

An Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒 Ѡ is cyclic if Ѡ = Ʀ𝑥 = 〈𝑥〉. 

Lemma 2.10[ 4, Ex(12). P 239]. 

1) Let 𝑉 is a submodule of an Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒 Ѡ with 𝑉 as a direct summand of Ѡ, then 𝐽 (
Ѡ

𝑉
) =

𝐽(Ѡ)+𝑉

𝑉
. 

2) An Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒 Ѡ is a 𝑠𝑒𝑚𝑖𝑠𝑖𝑚𝑝𝑙𝑒 if and only if for each submodule 𝑉 of Ѡ 𝑠𝑜𝑐 (
Ѡ

𝑉
) =

𝑠𝑜𝑐(Ѡ)+𝑉

𝑉
. 

Lemma 2.11[ 10, Ex12(5). P 242]. 
A submodule 𝑉 of 𝑎𝑛 Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒 Ѡ is 𝑚𝑎𝑥𝑖𝑚𝑎𝑙 and 𝑒𝑠𝑠𝑒𝑛𝑡𝑖𝑎𝑙 if and only if 𝑠𝑜𝑐(Ѡ) ⊆ 𝑉. 

Lemma 2.12[ 4, Lemma (2.3.15)]. 
“Let 𝐿, 𝑉 and 𝐷 are submodules of an Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒 Ѡ with 𝑉 ⊆ 𝐷, then (𝐿 + 𝑉) ∩ 𝐷 = (𝐿 ∩ 𝐷) +
𝑉 = (𝐿 ∩ 𝐷) + (𝑉 ∩ 𝐷).” 

Definition 2.13[11]. 

An Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒 Ѡ is multiplicatiion, if every submodule 𝑉 of Ѡ is of the form 𝑉 = 𝐼Ѡ for some 
𝑖𝑑𝑒𝑎𝑙 𝐼 of Ʀ. Equivalently, Ѡ is a multiplicatiion Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒 if every submodule 𝑉 of Ѡ of the 
form 𝑉 = [𝑉:Ʀ Ѡ]Ѡ.  

 



IHJPAS. 36(2)2023 

343 
 

 

Lemma 2.14[ 12, Prop. (2.3)]. 
Let Ѡ be a 𝑚ultiplication Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒. Then a submodule 𝑉 of Ѡ is a 2-Absorbing if and only if 𝑉 
Quasi-2-Absorbing submodule of Ѡ. 
 

3.  The Results  

In this part, we define Extend Nearly Pseudo Quasi-2-Absorbing submodule and characterize some 

of its fundamental properties using examples: 

Definition 3.1 

A 𝑝𝑟𝑜𝑝𝑒𝑟 submodule 𝑉 of an Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒 Ѡ is said to be Extend Nearly Pseudo Quasi-2-Absorbing 
( for short EXNPQ2AB ) submodule of Ѡ if whenever ɑɓ𝑐ӽ ∈ 𝑉, where ɑ,ɓ, 𝑐 ∈ Ʀ, ӽ ∈ Ѡ implies 
that either ɑ𝑐ӽ ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ)+𝐽(Ѡ) or ɓ𝑐ӽ ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or ɑɓӽ ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) +
𝐽(Ѡ).  
And an 𝑖𝑑𝑒𝑎𝑙 𝐼 of a ring Ʀ is called EXNPQ2AB ideal of Ʀ, if 𝐼 is an EXNPQ2AB Ʀ-𝑠𝑢𝑏𝑚𝑜𝑑𝑢𝑙𝑒 of 
an Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒 Ʀ. 

Remarks and Examples 3.2 

1. Every 2-Absorbing submodule of an Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒Ѡ is EXNPQ2AB submodule however, the 
opposite is not true. 

Proof. 

Let 𝑉 be a 2-Absorbing submodule of an Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒 Ѡ and ɑɓ𝑐ӽ ∈ 𝑉, for ɑ,ɓ, 𝑐 ∈ Ʀ, ӽ ∈ Ѡ. That 
is ɑɓ(𝑐ӽ) ∈ 𝑉. But 𝑉 is 2-Absorbing submodule of  Ѡ, then either ɑ(𝑐ӽ) ∈ 𝑉 or ɓ(𝑐ӽ) ∈ 𝑉 or 
ɑɓѠ ⊆ 𝑉. Thus, either ɑ𝑐ӽ ∈ 𝑉 ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or ɓ𝑐ӽ ∈ 𝑉 ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or 
ɑɓӽ ∈ 𝑉 ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) for all ӽ ∈ Ѡ, then either ɑ𝑐ӽ ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or ɓ𝑐ӽ ∈
𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or ɑɓӽ ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ). Hence, 𝑉 is EXNPQ2AB submodule of Ѡ. 

For the opposite, think about the following illustration: 

Let Ѡ = 𝑍48 , Ʀ = 𝑍 and the submodule 𝑉 = 〈16̅̅̅̅ 〉 is EXNPQ2AB submodule of Ѡ, 
since 𝑠𝑜𝑐(𝑍48) = 〈8̅〉  and 𝐽(𝑍48) = 〈6̅〉. That is for all ɑ, ɓ, 𝑐 ∈ 𝑍 and 𝑚 ∈  𝑍48 such that  ɑɓ𝑐𝑚 ∈
〈16̅̅̅̅ 〉 , implies that either ɑ𝑐𝑚 ∈ 〈16̅̅̅̅ 〉 + 𝑠𝑜𝑐(𝑍48) + 𝐽(𝑍48) = 〈16̅̅̅̅ 〉 + 〈8̅〉 + 〈6̅〉 = 〈2̅〉 or ɓ𝑐𝑚 ∈
〈8̅〉 + 𝑠𝑜𝑐(𝑍48) + 𝐽(𝑍48) = 〈16̅̅̅̅ 〉 + 〈8̅〉 + 〈6̅〉 = 〈2̅〉 or ɑɓ𝑚 ∈ 〈16̅̅̅̅ 〉 + 〈8̅〉 + 〈6̅〉 = 〈2̅〉. But 𝑉  is 
not 2-Absorbing, since  2.4. 2̅ ∈ 〈16̅̅̅̅ 〉, for 4,2 ∈ 𝑍 and  2̅ ∈ 𝑍48, implies that 4. 2̅ = 8̅ ∉ 〈16̅̅̅̅ 〉 and 
2. 2̅ = 4̅ ∉ 〈16̅̅̅̅ 〉 and 2.4 = 8 ∉ 16𝑍. 

2. Every Quasi-2-Absorbing submodule of an Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒 Ѡ is EXNPQ2AB submodule however, 
the opposite is not true. 

Proof.   

Let 𝑉 be a Quasi-2-Absorbing submodule of an Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒 Ѡ and ɑɓ𝑐ӽ ∈ 𝑉, for ɑ,ɓ, 𝑐 ∈ Ʀ, ӽ ∈
Ѡ. Since 𝑉 is Quasi-2-Absorbing submodule of  Ѡ, then either ɑ𝑐ӽ ∈ 𝑉 ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) 
or ɓ𝑐ӽ ∈ 𝑉 ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or ɑɓӽ ∈ 𝑉 ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ), hence either ɑ𝑐ӽ ∈ 𝑉 +
𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or ɓ𝑐ӽ ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or ɑɓӽ ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ). Therefore 𝑉 is 
EXNPQ2AB submodule of  Ѡ. 

For the opposite, think about the following illustration: 

Let Ѡ = 𝑍48, Ʀ = 𝑍 and the submodule 𝑉 = 〈12̅̅̅̅ 〉 is EXNPQ2AB submodule of Ѡ, 
since 𝑠𝑜𝑐(𝑍48) = 〈8̅〉  and 𝐽(𝑍48) = 〈6̅〉. That is for all ɑ, ɓ, 𝑐 ∈ 𝑍 and 𝑚 ∈  𝑍48 such that  ɑɓ𝑐𝑚 ∈



IHJPAS. 36(2)2023 

344 
 

〈12̅̅̅̅ 〉, implies that either ɑ𝑐𝑚 ∈ 〈12̅̅̅̅ 〉 + 𝑠𝑜𝑐(𝑍48) + 𝐽(𝑍48) = 〈12̅̅̅̅ 〉 + 〈8̅〉 + 〈6̅〉 = 〈2̅〉 or ɓ𝑐𝑚 ∈
〈12̅̅̅̅ 〉 + 𝑠𝑜𝑐(𝑍48) + 𝐽(𝑍48) = 〈12̅̅̅̅ 〉 + 〈8̅〉 + 〈6̅〉 = 〈2̅〉 or ɑɓ𝑚 ∈ 〈12̅̅̅̅ 〉 + 〈8̅〉 + 〈6̅〉 = 〈2̅〉. But 𝑉  is 
not Quasi-2-Absorbing, since  2.3.2. 1̅ ∈ 〈12̅̅̅̅ 〉, for 3,2 ∈ 𝑍 and  1̅ ∈ 𝑍48, implies that 2.2. 1̅ = 4̅ ∉
〈12̅̅̅̅ 〉 and 2.3. 1̅ = 6̅ ∉ 〈12̅̅̅̅ 〉. 

3. Every Nearly-2-Absorbing submodule of an Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒 Ѡ is EXNPQ2AB submodule however, 
the opposite is not true. 

Proof.  

Let 𝑉 be a Nearly-2-Absorbing submodule of an Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒 Ѡ and ɑɓ𝑐ӽ ∈ 𝑉, for ɑ,ɓ, 𝑐 ∈ Ʀ, ӽ ∈
Ѡ. That is ɑɓ(𝑐ӽ) ∈ 𝑉. But 𝑉 is Nearly-2-Absorbing submodule of  Ѡ, then either ɑ(𝑐ӽ) ∈ 𝑉 +
𝐽(Ѡ) or ɓ(𝑐ӽ) ∈ 𝑉 + 𝐽(Ѡ)  or ɑɓѠ ⊆ 𝑉 + 𝐽(Ѡ). Thus either ɑ𝑐ӽ ∈ 𝑉 + 𝐽(Ѡ)  ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) +
𝐽(Ѡ) or ɓ𝑐ӽ ∈ 𝑉 + 𝐽(Ѡ)  ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or ɑɓӽ ∈ 𝑉 + 𝐽(Ѡ)  ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) for 
all ӽ ∈ Ѡ, then either ɑ𝑐ӽ ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or ɓ𝑐ӽ ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or ɑɓӽ ∈ 𝑉 +
𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ). Hence 𝑉 is EXNPQ2AB submodule of Ѡ. 

For the opposite, think about the following illustration: 

Take a look at the 𝑍-𝑚𝑜𝑑𝑢𝑙𝑒 𝑍60 and the submodule 𝑉 = 〈30̅̅̅̅ 〉, we see that the only 
𝑒𝑠𝑠𝑒𝑛𝑡𝑖𝑎𝑙 submodules of  𝑍60 are 𝑍60 itself and the submodule 〈2̅〉, so that 𝑠𝑜𝑐(𝑍60) = 𝑍60 ∩
〈2̅〉 = 〈2̅〉. And the only 𝑚𝑎𝑥𝑖𝑚𝑎𝑙 submodules 〈2̅〉, 〈3̅〉 and 〈5̅〉. So that 𝐽(𝑍60) = 〈30̅̅̅̅ 〉, hence 〈30̅̅̅̅ 〉 
is EXNP-2-Absorbing submodule of 𝑍60, however Nearly-2-Absorbing submodule of 𝑍60, 
because 2.3. 5̅ ∈ 𝑉, for 2,3 ∈ 𝑍, 5̅ ∈ 𝑍60, implies that 2. 5̅ ∉ 𝑉 + 𝐽(𝑍60) = 〈30̅̅̅̅ 〉 + 〈30̅̅̅̅ 〉 = 〈30̅̅̅̅ 〉 
and 3. 5̅ ∉ 〈30̅̅̅̅ 〉 and 2.3 ∉ 30𝑍. 

4. Every Nearly Quasi-2-Absorbing submodule of an Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒 Ѡ is EXNPQ2AB submodule 
however, the opposite is not true. 

Proof.  

Clear. 

 

For the opposite, think about the following illustration: 

See the submodule 〈30̅̅̅̅ 〉 as the 𝑍-module 𝑍60. 

5. Every Pseudo-2-Absorbing submodule of an Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒 Ѡ is EXNPQ2AB submodule however, 
the opposite is not true. 

Proof.  

Let ɑɓ𝑐ӽ ∈ 𝑉, for ɑ,ɓ, 𝑐 ∈ Ʀ, ӽ ∈ Ѡ. That is ɑɓ(𝑐ӽ) ∈ 𝑉. But 𝑉 is Pseudo-2-Absorbing submodule 
of Ѡ, then either ɑ(𝑐ӽ) ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) or ɓ(𝑐ӽ) ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ)  or ɑɓѠ ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ). Thus 
either ɑ𝑐ӽ ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ)  ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or ɓ𝑐ӽ ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ)  ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) 
or ɑɓӽ ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ)  ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) for all ӽ ∈ Ѡ, then either ɑ𝑐ӽ ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) +
𝐽(Ѡ) or ɓ𝑐ӽ ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or ɑɓӽ ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ). Hence 𝑉 is EXNPQ2AB 
submodule of  Ѡ. 

For the opposite, think about the following illustration: 

Let Ѡ = 𝑍48, Ʀ = 𝑍 and the submodule 𝑉 = 〈8̅〉. Clear that 𝑉 is not Pseudo-2-Absorbing, but 𝑉 
is EXNPQ2AB submodule of Ѡ. 

6. Every Pseudo Quasi-2-Absorbing submodule of an Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒 Ѡ is EXNPQ2AB submodule 
however, the opposite is not true. 



IHJPAS. 36(2)2023 

345 
 

Proof.  

Direct. 

For the converse, see the example in (5). 

7. The intersection of two EXNPQ2AB submodules of an Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒 Ѡ need not to be 
EXNPQ2AB submodule of Ѡ, as the following example shows: 

Consider the 𝑍-𝑚𝑜𝑑𝑢𝑙𝑒 𝑍 and the submodules 3𝑍 and 4𝑍 are EXNPQ2AB submodules of 𝑍, but 
3𝑍 ∩ 4𝑍 = 12𝑍 is not EXNPQ2AB submodule of the 𝑍-𝑚𝑜𝑑𝑢𝑙𝑒 𝑍, since if 2.3.2.1 ∈ 12𝑍, but 
2.2.1 = 4 ∉ 12𝑍 + 𝑠𝑜𝑐(𝑍) + 𝐽(𝑍) and 3.2.1 = 6 ∉ 12𝑍 + 𝑠𝑜𝑐(𝑍) + 𝐽(𝑍) and 2.3.1 = 6 ∉=
12𝑍 + 𝑠𝑜𝑐(𝑍) + 𝐽(𝑍) (since 𝑠𝑜𝑐(𝑍) = (0) and 𝐽(𝑍) = 0). 

Proposition 3.3 

A 𝑝𝑟𝑜𝑝𝑒𝑟 submodule  𝑉 of  Ѡ  is EXNPQ2AB submodule of  Ѡ  if and only if for any  ɑ , ɓ ∈  Ʀ 
and 𝑥 ∈ Ѡ such that ɑɓ𝑥 ∉ 𝑉 +  𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ). Then [𝑉 :Ʀ ɑɓ𝑥 ] ⊆ [𝑉 +  𝑠𝑜𝑐(Ѡ) +
𝐽(Ѡ):Ʀ ɑ𝑥 ] ∪ [𝑉 +  𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ):Ʀ  ɓ𝑥]. 

Proof.   

(⟹) Let 𝑉 be EXNPQ2AB submodule of Ѡ and 𝑡 ∈ [𝑉 :Ʀ ɑɓ𝑥] , then ɑɓ𝑡𝑥 ∈ 𝑉. Since 𝑉 is 
EXNPQ2AB submodule of  Ѡ and ɑɓ𝑥 ∉ 𝑉 +  𝑠𝑜𝑐 (Ѡ) + 𝐽(Ѡ), then either ɑ𝑡𝑥 ∈ 𝑉 +
𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or ɓ𝑡𝑥 ∈ 𝑉 +  𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ). Thus either  𝑡 ∈ [𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ):Ʀ ɑ𝑥] or 
𝑡 ∈[𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) :Ʀ  ɓ𝑥]. Hence, 𝑡 ∈ [𝑉 +  𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ):Ʀ ɑ𝑥] ∪ [𝑉 +  𝑠𝑜𝑐 (Ѡ) +
𝐽(Ѡ):Ʀ ɓ𝑥]. Then we get [𝑉 :Ʀ ɑɓ𝑥] ⊆ [𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ):Ʀ ɑ𝑥 ] ∪ [𝑉 +  𝑠𝑜𝑐(Ѡ) +
𝐽(Ѡ):Ʀ  ɓ𝑥]. 

(⟸) Let ɑɓ𝑐𝑥 ∈ 𝑉 for ɑ, ɓ, 𝑐 ∈  Ʀ, 𝑥 ∈  Ѡ  and let ɑɓ𝑥 ∉ 𝑉 +  𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ). Since ɑɓ𝑐𝑥 ∈
𝑉, then 𝑐 ∈ [𝑉 :Ʀ ɑɓ𝑥 ] ⊆ [𝑉 +  𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ):Ʀ   ɑ𝑥] ∪ [ 𝑉 +  𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ):Ʀ  ɓ𝑥]. It 
follows that either 𝑐 ∈  [ 𝑉 +  𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ):Ʀ   ɑ𝑥] or 𝑐 ∈ [ 𝑉 +  𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ):Ʀ   ɓ𝑥]. That 
is either ɑ𝑐𝑥 ∈ 𝑉 +  𝑠𝑜𝑐 (Ѡ) + 𝐽(Ѡ) or  ɓ𝑐𝑥 ∈ 𝑉 +  𝑠𝑜𝑐 (Ѡ) + 𝐽(Ѡ). Therefore, 𝑉 
EXNPQ2AB submodule of  Ѡ. 

Proposition 3.4 

A 𝑝𝑟𝑜𝑝𝑒𝑟 submodule  𝑉  of Ѡ  is EXNPQ2AB submodule of Ѡ if and only if  ɑɓ𝑐𝕃 ⊆ 𝑉, for  ɑ, 
ɓ, 𝑐 ∈  Ʀ  and 𝕃  is a submodule of Ѡ, implies that either ɑ𝑐𝕃 ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or ɓ𝑐𝕃 ⊆
𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or  ɑɓ𝕃 ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ). 

Proof.   

(⟹) Let  𝑉  be EXNPQ2AB submodule of Ѡ and  ɑɓ𝑐𝕃 ⊆ 𝑉 , for  ɑ, ɓ, 𝑐 ∈  Ʀ  and 𝕃 is a 
submodule of Ѡ. Suppose that  ɑɓ𝕃 ⊈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) , ɑ𝑐𝕃 ⊈  𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) 
and ɓ𝑐𝕃 ⊈  𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ). Then there is  𝑒1, 𝑒2, 𝑒3 ∈ 𝕃 such that ɑɓ𝑒1 ∉  𝑉 + 𝑠𝑜𝑐(Ѡ) +
𝐽(Ѡ) ,ɑ𝑐𝑒2 ∉  𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ)    and   ɓ𝑐𝑒3 ∉  𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) . Now, ɑɓ𝑐𝑒1 ∈ 𝑉  and 
since 𝑉 is EXNPQ2AB submodule of Ѡ with ɑɓ𝑒1 ∉  𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ), then either ɓ𝑐𝑒1 ∈
𝑉 +  𝑠𝑜𝑐 (Ѡ) + 𝐽(Ѡ) or ɑ𝑐𝑒1 ∈ 𝑉 +  𝑠𝑜𝑐 (Ѡ) + 𝐽(Ѡ). Also since ɑɓ𝑐𝑒2 ∈ 𝑉  and  ɑ𝑐𝑒2 ∉  𝑉 +
𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ), then either ɓ𝑐𝑒2 ∈ 𝑉 +  𝑠𝑜𝑐 (Ѡ) + 𝐽(Ѡ) or ɑɓ𝑒2 ∈ 𝑉 +  𝑠𝑜𝑐 (Ѡ) + 𝐽(Ѡ). 
Again ɑɓ𝑐𝑒3 ∈ 𝑉  and since 𝑉 is EXNPQ2AB submodule of Ѡ with ɓ𝑐𝑒3 ∉  𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ), 
then either ɑ𝑐𝑒3 ∈ 𝑉 +  𝑠𝑜𝑐 (Ѡ) + 𝐽(Ѡ) or ɑɓ𝑒3 ∈ 𝑉 +  𝑠𝑜𝑐 (Ѡ) + 𝐽(Ѡ). Now, ɑɓ𝑐 ( 𝑒1  +
 𝑒2 + 𝑒3 ) ∈ 𝑉 and 𝑉 is EXNPQ2AB submodule of Ѡ, implies that either  ɑɓ(𝑒1  +  𝑒2 + 𝑒3) ∈
𝑉 +  𝑠𝑜𝑐 (Ѡ) + 𝐽(Ѡ) or ɑ𝑐(𝑒1  +  𝑒2 + 𝑒3) ∈ 𝑉 +  𝑠𝑜𝑐 (Ѡ) + 𝐽(Ѡ) or ɓ𝑐(𝑒1  +  𝑒2 + 𝑒3) ∈
𝑉 +  𝑠𝑜𝑐 (Ѡ) + 𝐽(Ѡ). If ɑɓ(𝑒1  +  𝑒2 + 𝑒3) = ɑɓ𝑒1 + ɑɓ𝑒2 + ɑɓ𝑒3 ∈ 𝑉 +  𝑠𝑜𝑐 (Ѡ) + 𝐽(Ѡ). 
But ɑɓ𝑒2 ∈ 𝑉 +  𝑠𝑜𝑐 (Ѡ) + 𝐽(Ѡ) and   ɑɓ𝑒3 ∈ 𝑉 +  𝑠𝑜𝑐 (Ѡ) + 𝐽(Ѡ)    , then  ɑɓ𝑒1 ∈ 𝑉 +
 𝑠𝑜𝑐 (Ѡ) + 𝐽(Ѡ), which is incongruent. If ɑ𝑐(𝑒1  +  𝑒2 + +𝑒3) = ɑ𝑐𝑒1 + ɑ𝑐𝑒2 + ɑ𝑐𝑒3 ∈ 𝑉 +
 𝑠𝑜𝑐 (Ѡ) + 𝐽(Ѡ). But ɑ𝑐𝑒1 ∈ 𝑉 +  𝑠𝑜𝑐 (Ѡ) + 𝐽(Ѡ) and   ɑ𝑐𝑒3 ∈ 𝑉 +  𝑠𝑜𝑐 (Ѡ) + 𝐽(Ѡ), then  



IHJPAS. 36(2)2023 

346 
 

ɑ𝑐𝑒2 ∈ 𝑉 +  𝑠𝑜𝑐 (Ѡ) + 𝐽(Ѡ) which is contradiction.   If  ɓ𝑐(𝑒1  +  𝑒2 + +𝑒3) = ɓ𝑐𝑒1 + ɓ𝑐𝑒2 +
ɓ𝑐𝑒3 ∈ 𝑉 +  𝑠𝑜𝑐 (Ѡ) + 𝐽(Ѡ). But ɓ𝑐𝑒1 ∈ 𝑉 +  𝑠𝑜𝑐 (Ѡ) + 𝐽(Ѡ) and   ɓ𝑐𝑒2 ∈ 𝑉 +  𝑠𝑜𝑐 (Ѡ) +
𝐽(Ѡ), then  ɓ𝑐𝑒3 ∈ 𝑉 +  𝑠𝑜𝑐 (Ѡ) + 𝐽(Ѡ) which is contradiction. Hence ɑ𝑐𝕃 ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) +
𝐽(Ѡ) or ɓ𝑐𝕃 ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or ɑɓ𝕃 ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ). 

(⟸) Let   ɑɓ𝑐𝑛 ∈ 𝑉  for  ɑ, ɓ, 𝑐 ∈  Ʀ , 𝑛 ∈  Ѡ, then  ɑɓ𝑐(𝑛) ⊆ 𝑉 , hence  by  hypothesis either  
ɑ𝑐(𝑛) ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ)  or   ɓ𝑐(𝑛) ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ)  or  ɑɓ(𝑛) ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) +
𝐽(Ѡ). That  is  either ɑ𝑐𝑛 ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or ɓ𝑐𝑛 ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or ɑɓ𝑛 ∈ 𝑉 +
𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ). Therefore  𝑉  is EXNPQ2AB submodule of Ѡ. 

Proposition 3.5 

Let Ѡ be 𝑚𝑜𝑑𝑢𝑙𝑒 and 𝑉 be a 𝑝𝑟𝑜𝑝𝑒𝑟 submodule of Ѡ. Then 𝑉 is EXNPQ2AB submodule of Ѡ 
if and only if for every submodule 𝐴 of Ѡ and for every ideals 𝐼1, 𝐼2, 𝐼3 of Ʀ such that 𝐼1𝐼2𝐼3𝐴 ⊆ 𝑉, 
implies that either 𝐼1𝐼2𝐴 ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or 𝐼1𝐼3𝐴 ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ)  or 𝐼2𝐼3𝐴 ⊆ 𝑉 +
𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ). 

Proof.   

(⟹) Let 𝐼1𝐼2𝐼3𝐴 ⊆ 𝑉, where 𝐼1, 𝐼2, 𝐼3 are ideals of Ʀ and 𝐴 is a submodule of Ѡ, with 𝐼1𝐼2𝐴  ⊈
 [ 𝑉 +  𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) :Ʀ  Ѡ ]. To demonstrate that 𝐼1𝐼3𝐴 ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or 𝐼2𝐼3𝐴 ⊆
𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ). Suppose that  𝐼1𝐼3𝐴 ⊈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) and  𝐼2𝐼3𝐴 ⊈ 𝑉 + 𝑠𝑜𝑐(Ѡ) +
𝐽(Ѡ), that is there exist  𝑎1, 𝑎2, 𝑎3 ∈ 𝐴 and a nonzero 𝑟 ∈ 𝐼1, 𝑠 ∈ 𝐼2 and 𝑡 ∈ 𝐼3 such that 𝑟𝑠𝑎1 ∉
𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) and 𝑟𝑡𝑎2 ∉ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) and 𝑠𝑡𝑎3 ∉ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ). Now, 
𝑟𝑠𝑡𝑎1 ∈ 𝑉 and 𝑟𝑠𝑎1 ∉ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ), implies that either 𝑟𝑡𝑎1 ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or 
𝑠𝑡𝑎1 ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ). Also 𝑟𝑠𝑡𝑎2 ∈ 𝑉 and 𝑟𝑡𝑎2 ∉ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ), implies that either 
𝑟𝑠𝑎2 ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or 𝑠𝑡𝑎2 ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ). Again, 𝑟𝑠𝑡𝑎3 ∈ 𝑉 and 𝑠𝑡𝑎3 ∉ 𝑉 +
𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ), implies that either 𝑟𝑡𝑎3 ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or  𝑟𝑠𝑎3 ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ). 
Now, 𝑟𝑠𝑡(𝑎1 + 𝑎2 + 𝑎3) ∈ 𝑉 and 𝑉 is EXNPQ2AB, then either 𝑟𝑠(𝑎1 + 𝑎2 + 𝑎3) ∈ 𝑉 +
𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or 𝑟𝑡(𝑎1 + 𝑎2 + 𝑎3) ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or 𝑠𝑡(𝑎1 + 𝑎2 + 𝑎3) ∈ 𝑉 +
𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ). If 𝑟𝑠(𝑎1 + 𝑎2 + 𝑎3) = 𝑟𝑠𝑎1 + 𝑟𝑠𝑎2 + 𝑟𝑠𝑎3 ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) and 𝑟𝑠𝑎2, 
𝑟𝑠𝑎3 ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ), hence 𝑟𝑠𝑎1 ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) which is a contradiction. If 
𝑟𝑡(𝑎1 + 𝑎2 + 𝑎3) = 𝑟𝑡𝑎1 + 𝑟𝑡𝑎2 + 𝑟𝑡𝑎3 ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) and 𝑟𝑡𝑎1, 𝑟𝑡𝑎3 ∈ 𝑉 +
𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ), hence 𝑟𝑡𝑎2 ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) which is a contradiction. If 𝑠𝑡(𝑎1 + 𝑎2 +
𝑎3) = 𝑠𝑡𝑎1 + 𝑠𝑡𝑎2 + 𝑠𝑡𝑎3 ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) and 𝑠𝑡𝑎1, 𝑠𝑡𝑎2 ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ), hence 
𝑠𝑡𝑎3 ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) which is a contradiction. Thus either 𝐼1𝐼2𝐴 ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) 
or 𝐼1𝐼3𝐴 ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ)  or 𝐼2𝐼3𝐴 ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ). 

(⟸) Clear. 

Proposition 3.6 

Let Ѡ be 𝑚𝑜𝑑𝑢𝑙𝑒 and 𝑉 be a 𝑝𝑟𝑜𝑝𝑒𝑟 submodule of Ѡ. Then 𝑉 is EXNPQ2AB submodule of Ѡ 
if and only if for any 𝑟, 𝑠 ∈ Ʀ and 𝐼of Ʀ and 𝑥 ∈ Ѡ with 𝑟𝑠𝐼𝑥 ⊆ 𝑉  implies that either 𝑟𝑠𝑥 ∈ 𝑉 +
𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or 𝑟𝐼𝑥 ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ)  or 𝑠𝐼𝑥 ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ).  

Proof.   

(⟹) Let 𝑟𝑠𝐼𝑥 ⊆ 𝑉 for 𝑟, 𝑠 ∈ Ʀ and 𝐼 is an ideal of Ʀ and 𝑥 ∈ Ѡ, it follows that 𝐼 ⊆ [𝑉:Ʀ 𝑟𝑠𝑥]. If 
𝑟𝑠𝑥 ∈ 𝑉 ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ), hence 𝑟𝑠𝑥 ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ), then we are done. Suppose 
that 𝑟𝑠𝑥 ∉ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ), then by Proposition 3.3 [ 𝑉 :Ʀ 𝑟𝑠𝑥 ] ⊆ [ 𝑉 +  𝑠𝑜𝑐 (Ѡ) +
𝐽(Ѡ):Ʀ 𝑟𝑥 ] ∪ [ 𝑉 +  𝑠𝑜𝑐 (Ѡ) + 𝐽(Ѡ):Ʀ  𝑠𝑥 ]. But 𝑟𝑠𝐼𝑥 ⊆ 𝑉, then 𝐼 ⊆ [𝑉:Ʀ 𝑟𝑠𝑥] ⊆ [ 𝑉 +
 𝑠𝑜𝑐 (Ѡ) + 𝐽(Ѡ):Ʀ 𝑟𝑥 ] ∪ [ 𝑉 +  𝑠𝑜𝑐 (Ѡ) + 𝐽(Ѡ):Ʀ  𝑠𝑥 ], hence 𝐼 ⊆ [ 𝑉 + 𝑠𝑜𝑐 (Ѡ) +
𝐽(Ѡ):Ʀ 𝑟𝑥] ∪ [𝑉 + 𝑠𝑜𝑐 (Ѡ) + 𝐽(Ѡ):Ʀ 𝑠𝑥], it follows that either 𝐼 ⊆ [𝑉 + 𝑠𝑜𝑐 (Ѡ) + 𝐽(Ѡ):Ʀ 𝑟𝑥] 
or 𝐼 ⊆ [𝑉 +  𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ):Ʀ 𝑠𝑥], thus either 𝑟𝐼𝑥 ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ)  or 𝑠𝐼𝑥 ⊆ 𝑉 +
𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ).  



IHJPAS. 36(2)2023 

347 
 

(⟸) Let 𝑟𝑠𝑡𝑥 ∈ 𝑉 for 𝑟, 𝑠, 𝑡 ∈ Ʀ and 𝑥 ∈ Ѡ, that is 𝑟𝑠〈𝑡〉𝑥 ⊆ 𝑉. It follows by hypothesis either 
𝑟𝑠𝑥 ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or 𝑟〈𝑡〉𝑥 ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or 𝑠〈𝑡〉𝑥 ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ). 
Hence either 𝑟𝑠𝑥 ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or 𝑟𝑡𝑥 ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or 𝑠𝑡𝑥 ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) +
𝐽(Ѡ). Therefore 𝑉 is EXNPQ2AB submodule of Ѡ. 

From the Proposition 3.5 and Proposition 3.6 we get the following corollaries. 

Corollary 3.7 

Let Ѡ be an Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒 and 𝑉 be a 𝑝𝑟𝑜𝑝𝑒𝑟 submodule of  Ѡ. Then 𝑉 is EXNPQ2AB submodule 
of Ѡ if and only if for each 𝑟 ∈ Ʀ, 𝑥 ∈ Ѡ and every ideals 𝐼, 𝐽 of Ʀ with 𝑟𝐼𝐽𝑥 ⊆ 𝑉, implies that 
either 𝑟𝐼𝑥 ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or 𝑟𝐽𝑥 ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or 𝐼𝐽𝑥 ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ).  

Corollary 3.8 

Let Ѡ be an Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒 and 𝑉 be a 𝑝𝑟𝑜𝑝𝑒𝑟 submodule of Ѡ. Then 𝑉 is EXNPQ2AB submodule 
of Ѡ if and only if for every ideals 𝐼1, 𝐼2, 𝐼3 of Ʀ and 𝑥 ∈ Ѡ such that 𝐼1𝐼2𝐼3𝑥 ⊆ 𝑉  implies that 
either 𝐼1𝐼2𝑥 ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or 𝐼1𝐼3𝑥 ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ)  or 𝐼2𝐼3𝑥 ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) +
𝐽(Ѡ).  

Corollary 3.9 

Let Ѡ be an Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒 and 𝑉 be a 𝑝𝑟𝑜𝑝𝑒𝑟 submodule of Ѡ. Then 𝑉 is EXNPQ2AB submodule 
of Ѡ if and only if for any 𝑟, 𝑠 ∈ Ʀ and any ideal 𝐼 of Ʀ and every submodule 𝐴 of Ѡ with 𝑟𝑠𝐼𝐴 ⊆
𝑉  implies that either 𝑟𝑠𝐴 ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or 𝑟𝐼𝐴 ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ)  or 𝑠𝐼𝐴 ⊆ 𝑉 +
𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ).  

Corollary 3.10 

Let Ѡ be an Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒 and 𝑉 be a 𝑝𝑟𝑜𝑝𝑒𝑟 submodule of Ѡ. Then 𝑉 is EXNPQ2AB submodule 
of Ѡ if and only if for each 𝑟 ∈ Ʀ and any ideals 𝐼, 𝐽 of Ʀ and every submodule 𝐴 of Ѡ with 𝑟𝐼𝐽𝐴 ⊆
𝑉  implies that either 𝑟𝐼𝐴 ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or 𝑟𝐽𝐴 ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ)  or 𝐼𝐽𝐴 ⊆ 𝑉 +
𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ).  
 

Proposition 3.11 

Let 𝑉  be EXNPQ2AB  submodule of   an Ʀ- 𝑚𝑜𝑑𝑢𝑙𝑒 Ѡ and   L  is a submodule  of  Ѡ  with  L 

⊆  𝑉 , then 
   𝑉   

 𝐿  
 is EXNPQ2AB submodule  of  an Ʀ-module   

Ѡ   

𝐿  
 . 

Proof.   

Let  𝑉  be EXNPQ2AB submodule of Ѡ and  ɑ𝐼𝐽 ( 𝑒 +  𝐿 )  =  ɑ𝐼𝐽𝑒 +   𝐿  ⊆  
   𝑉   

  𝐿   
 for ɑ ∈  Ʀ, 𝐼, 𝐽 

are ideals of Ʀ and  𝑒 +  𝐿  ∈ 
Ѡ   

𝐿  
 , 𝑒 ∈ Ѡ , implies  that  ɑ𝐼𝐽𝑒 ⊆ 𝑉.  Since 𝑉  is EXNPQ2AB 

submodule of Ѡ,  then by Corollary 3.7 either  ɑ𝐼𝑒 ⊆ 𝑉 + 𝑠𝑜𝑐 (Ѡ ) + 𝐽(Ѡ)  or  ɑ𝐽𝑒 ⊆ 𝑉 
+ 𝑠𝑜𝑐 (Ѡ ) + 𝐽(Ѡ) or  𝐼𝐽𝑒 ⊆ 𝑉 + 𝑠𝑜𝑐 (Ѡ ) + 𝐽(Ѡ). It follows,  either ɑ𝐼( 𝑒 +  𝐿 )  ⊆
𝑉 + 𝑠𝑜𝑐 (Ѡ )+𝐽(Ѡ)

𝐿 
⊆

   𝑉   

 𝐿  
+

𝑉 + 𝑠𝑜𝑐 (Ѡ )+𝐽(Ѡ)

𝐿 
⊆

   𝑉   

 𝐿  
+ 𝑠𝑜𝑐 (

Ѡ   

𝐿  
) + 𝐽(

Ѡ   

𝐿  
) or ɑ𝐽( 𝑒 +  𝐿 ) ⊆

   𝑉   

 𝐿  
+

𝑠𝑜𝑐 (
Ѡ   

𝐿  
) + 𝐽(

Ѡ   

𝐿  
) or 𝐼𝐽( 𝑒 +  𝐿 )  ⊆

𝑉 + 𝑠𝑜𝑐 (Ѡ )+𝐽(Ѡ)

𝐿 
⊆

   𝑉   

 𝐿  
+

𝑉 + 𝑠𝑜𝑐 (Ѡ )+𝐽(Ѡ)

𝐿 
⊆

   𝑉   

 𝐿  
+

𝑠𝑜𝑐 (
Ѡ   

𝐿  
) + 𝐽(

Ѡ   

𝐿  
), that is either  ɑ𝐼( 𝑒 +  𝐿 ) ⊆

   𝑉   

  𝐿   
+ 𝑠𝑜𝑐 (

Ѡ   

𝐿  
) + 𝐽(

Ѡ   

𝐿  
) or ɑ𝐽( 𝑒 +  𝐿 ) ⊆

   𝑉   

  𝐿   
+

𝑠𝑜𝑐 (
Ѡ   

𝐿  
) + 𝐽(

Ѡ   

𝐿  
) or   𝐼𝐽 ⊆

   𝑉   

  𝐿   
+ 𝑠𝑜𝑐 (

Ѡ   

𝐿  
) + 𝐽(

Ѡ   

𝐿  
). Hence by Corollary 3.7 

   𝑉   

 𝐿  
 is EXNPQ2AB 

submodule of   
Ѡ   

𝐿  
. 

 

 

 

 



IHJPAS. 36(2)2023 

348 
 

Proposition 3.12 

Let Ѡ is a 𝑠𝑒𝑚𝑖 𝑠𝑖𝑚𝑝𝑙𝑒 Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒 𝑉 and Ң are submdules for Ѡ such that Ң ⊆  𝑉, and 𝑉 is a 

𝑝𝑟𝑜𝑝𝑒𝑟 submodule of Ѡ. If Ң and 
𝑉

Ң
 are EXNPQ2AB submodules of Ѡ and 

Ѡ

Ң
 respectively, then 

𝑉 is EXNPQ2AB submodules of Ѡ. 

Proof.   

Suppose Ң and 
𝑉

Ң
 are EXNPQ2AB submodules for Ѡ and 

Ѡ

Ң
 respectively, and let  𝐼1𝐼2𝐼3𝑚 ⊆ 𝑉, for 

𝐼1, 𝐼2, 𝐼3 are ideals of Ɍ, 𝑚 ∈ Ѡ. So 𝐼1𝐼2𝐼3(𝑚 + Ң) = 𝐼1𝐼2𝐼3𝑚 + Ң ⊆
𝑉

Ң
 . If 𝐼1𝐼2𝐼3𝑚 ⊆ Ң and Ң is 

EXNPQ2AB submodules of Ѡ, implies that by Corollary 3.8 either 𝐼1𝐼2𝑚 ⊆ Ң + (𝑠𝑜𝑐(Ѡ) +
𝐽(Ѡ)) ⊆  𝑉 + (𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ)) or 𝐼1𝐼3𝑚 ⊆ Ң + (𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ)) ⊆   𝑉 + (𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ)) 
or 𝐼2𝐼3𝑚 ⊆ Ң + (𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ)) ⊆  𝑉 + (𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ)), hence 𝑉 is EXNPQ2AB 

submodules for Ѡ. So, we may assume that 𝐼1𝐼2𝐼3𝑚 ⊈ Ң. It follows that 𝐼1𝐼2𝐼3(𝑚 + Ң) ⊆
Ŋ

Ң
 , but 

𝑉

Ң
 is EXNPQ2AB submodules of  

Ѡ

Ң
  , again by Corollary 3.8 either 𝐼1𝐼2(𝑚 + Ң) ⊆  

𝑉

Ң
+

(𝑠𝑜𝑐 (
Ѡ

Ң
 ) + 𝐽 (

Ѡ

Ң
 )) or 𝐼1𝐼3(𝑚 + Ң) ⊆  

𝑉

Ң
+ (𝑠𝑜𝑐 (

Ѡ

Ң
 ) + 𝐽 (

Ѡ

Ң
 )) or 𝐼2𝐼3(𝑚 + Ң) ⊆  

𝑉

Ң
+

(𝑠𝑜𝑐 (
Ѡ

Ң
 ) + 𝐽 (

Ѡ

Ң
 )). Since Ѡ is a 𝑠𝑒𝑚𝑖 𝑠𝑖𝑚𝑝𝑙𝑒 then by Lemma 2.10 either 𝐼1𝐼2(𝑚 + Ң) ⊆  

𝑉

Ң
+

Ң+𝑠𝑜𝑐(Ѡ)

Ң
+

Ң+𝐽(Ѡ)

Ң
 or 𝐼1𝐼3(𝑚 + Ң) ⊆  

𝑉

Ң
+

Ң+𝑠𝑜𝑐(Ѡ)

Ң
+

Ң+𝐽(Ѡ)

Ң
 or 𝐼2𝐼3(𝑚 + Ң)  ⊆  

𝑉

Ң
+

Ң+𝑠𝑜𝑐(Ѡ)

Ң
+

Ң+𝐽(Ѡ)

Ң
. But Ң ⊆ 𝑉, it follows that Ң + 𝑠𝑜𝑐(Ѡ) ⊆  𝑉 + 𝑠𝑜𝑐(Ѡ) and Ң + 𝐽(Ѡ) ⊆  𝑉 + 𝐽(Ѡ), 

hence 
𝑉

Ң
+

Ң+𝑠𝑜𝑐(Ѡ)

Ң
+

Ң+𝐽(Ѡ)

Ң
⊆  

𝑉

Ң
+

𝑉+𝑠𝑜𝑐(Ѡ)

Ң
+

𝑉+𝐽(Ѡ)

Ң
=

𝑉+𝑠𝑜𝑐(Ѡ)+𝐽(Ѡ)

Ң
. Thus either 𝐼1𝐼2(𝑚 +

Ң) ⊆  
𝑉+(𝑠𝑜𝑐(Ѡ)+𝐽(Ѡ))

Ң
 or 𝐼1𝐼3(𝑚 + Ң) ⊆

𝑉+(𝑠𝑜𝑐(Ѡ)+𝐽(Ѡ))

Ң
  or 𝐼1𝐼3(𝑚 + Ң) ⊆  

𝑉+(𝑠𝑜𝑐(Ѡ)+𝐽(Ѡ))

Ң
, it 

follows that  either 𝐼1𝐼2𝑚 ⊆ 𝑉 + (𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ)) or 𝐼1𝐼3𝑚 ⊆ 𝑉 + (𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ))or 𝐼2𝐼3𝑚 ⊆
 𝑉 + (𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ)). Hence by Corollary 3.8 𝑉 is EXNPQ2AB submodules of Ѡ. 

According to the following proposition, under specific circumstances, the intersection of two 

EXNPQ2AB submodules is an EXNPQ2AB submodule. 

Proposition 3.13 

Let Ѡ be an Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒 either 𝐸 or 𝑉 is 𝑚𝑎𝑥𝑖𝑚𝑎𝑙 𝑒𝑠𝑠𝑒𝑛𝑡𝑖𝑎𝑙 submodule of Ѡ and 𝐸 not contained 
in 𝑉. If 𝐸 and 𝑉 are EXNPQ2AB submodules of Ѡ, then 𝑉 ∩ 𝐸 is EXNPQ2AB submodule of Ѡ.  

Proof.   

Since 𝐸 not contained in 𝑉, then 𝑉 ∩ 𝐸 is a 𝑝𝑟𝑜𝑝𝑒𝑟 submodule of  𝑉 and since 𝑉  is a 𝑝𝑟𝑜𝑝𝑒𝑟 
submodule of Ѡ, hence 𝑉 ∩ 𝐸 is a 𝑝𝑟𝑜𝑝𝑒𝑟 submodule of Ѡ. Now, let 𝑟𝑠𝐼𝑥 ⊆ 𝐸 ∩ 𝑉, for 𝑟, 𝑠 ∈ Ʀ, 
𝑥 ∈ Ѡ and 𝐼 is an ideal of Ʀ, it follows that 𝑟𝑠𝐼𝑥 ⊆ 𝐸 and 𝑟𝑠𝐼𝑥 ⊆ 𝑉. But both 𝐸 and 𝑉 are 
EXNPQ2AB submodules of Ѡ, then by Proposition 3.6 we have either 𝑟𝑠𝑥 ∈ 𝐸 + 𝑠𝑜𝑐(Ѡ) +
𝐽(Ѡ) or 𝑟𝐼𝑥 ⊆ 𝐸 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ)  or 𝑠𝐼𝑥 ⊆ 𝐸 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ)  and 𝑟𝑠𝑥 ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) +
𝐽(Ѡ) or 𝑟𝐼𝑥 ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or 𝑠𝐼𝑥 ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ). Thus either 𝑟𝑠𝑥 ∈ (𝐸 +
𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ)) ∩ (𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ)) or 𝑟𝐼𝑥 ⊆ (𝐸 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ)) ∩ (𝑉 + 𝑠𝑜𝑐(Ѡ) +
𝐽(Ѡ))  or 𝑠𝐼𝑥 ⊆ (𝐸 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ)) ∩ (𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ)). Since either 𝐸 or 𝑉 is 
𝑚𝑎𝑥𝑖𝑚𝑎𝑙 𝑒𝑠𝑠𝑒𝑛𝑡𝑖𝑎𝑙 submodule of Ѡ, then either 𝑠𝑜𝑐(Ѡ) ⊆ 𝐸  or 𝑠𝑜𝑐(Ѡ) ⊆ 𝑉. Suppose that 𝐸 
is 𝑚𝑎𝑥𝑖𝑚𝑎𝑙 𝑒𝑠𝑠𝑒𝑛𝑡𝑖𝑎𝑙 submodule of Ѡ, so that by Lemma 2.11 𝑠𝑜𝑐(Ѡ) ⊆ 𝐸 and since 𝐸 is 
𝑚𝑎𝑥𝑖𝑚𝑎𝑙 submodule of Ѡ, then 𝐽(Ѡ) ⊆ 𝐸. It follows’ that 𝐸 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) = 𝐸. Hence 
either 𝑟𝑠𝑥 ∈ 𝐸 ∩ (𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ)) or 𝑟𝐼𝑥 ⊆ 𝐸 ∩ (𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ)) or 𝑠𝐼𝑥 ⊆ 𝐸 ∩ (𝑉 +
𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ)). Therefore by Lemma 2.12 we get either 𝑟𝑠𝑥 ∈ (𝐸 ∩ 𝑉) + (𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ)) or 



IHJPAS. 36(2)2023 

349 
 

𝑟𝐼𝑥 ⊆ (𝐸 ∩ 𝑉) + (𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ)) or 𝑠𝐼𝑥 ⊆ (𝐸 ∩ 𝑉) + (𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ)). Hence by 
Proposition 3.6 𝑉 ∩ 𝐸 is EXNPQ2AB submodule of  Ѡ.  

Proposition 3.14 

Let Ѡ be an Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒 with 𝑠𝑜𝑐(Ѡ) is Quasi-2-Absorbing submodule of  Ѡ . If   𝑉 ⊂ Ѡ such 
that  𝑉 ⊆  𝑠𝑜𝑐(Ѡ) , then   𝑉  is   EXNPQ2AB submodule of  Ѡ. 

Proof.   

Let 𝐼1𝐼2𝐼3𝐴 ⊆ 𝑉 , for 𝐼1, 𝐼2, 𝐼3 are ideals of Ʀ and 𝐴  is a submodule  of  Ѡ . Since  𝑉 ⊆ 𝑠𝑜𝑐 (Ѡ), 
it  follows that 𝐼1𝐼2𝐼3𝐴 ⊆ 𝑠𝑜𝑐 (Ѡ). But 𝑠𝑜𝑐(Ѡ)  is Quasi-2-Absorbing subomdule of Ѡ, then 
either 𝐼1𝐼3𝐴 ⊆ 𝑠𝑜𝑐 (Ѡ) ⊆  𝑉 +  𝑠𝑜𝑐 (Ѡ) + 𝐽(Ѡ) or 𝐼2𝐼3𝐴 ⊆ 𝑠𝑜𝑐 (Ѡ) ⊆  𝑉 +  𝑠𝑜𝑐 (Ѡ) + 𝐽(Ѡ) 
or  𝐼1𝐼2𝐴 ⊆ 𝑠𝑜𝑐 (Ѡ) ⊆  𝑉 + 𝑠𝑜𝑐 (Ѡ) + 𝐽(Ѡ). That is either  𝐼1𝐼3𝐴 ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) +  𝐽(Ѡ) or 
𝐼2𝐼3𝐴 ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or 𝐼1𝐼2𝐴 ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) +  𝐽(Ѡ).Therefore by Proposition 3.5 𝑉 
is EXNPQ2AB submodule of Ѡ. 

Proposition 3.14 

Let Ѡ be an Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒 with 𝐽(Ѡ) is Quasi-2-Absorbing submodule of  Ѡ . If   𝑉 ⊂ Ѡ such that  
𝑉 ⊆  𝐽(Ѡ) , then   𝑉  is   EXNPQ2AB submodule of  Ѡ. 

Proof.   

Let  ɑɓ𝑐𝐴 ⊆ 𝑉 , for ɑ , ɓ, 𝑐 ∈  Ʀ and 𝐴  is a submodule  of  Ѡ . Since  𝑉 ⊆  𝐽(Ѡ), it  follows that 
ɑɓ𝐴 ⊆ 𝐽(Ѡ). But 𝐽(Ѡ)  is Quasi-2-Absorbing subomdule of Ѡ , then either  ɑ𝑐𝐴 ⊆ 𝐽(Ѡ) ⊆  𝑉 +
 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or ɓ𝑐𝐴 ⊆ 𝐽(Ѡ) ⊆  𝑉 +  𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or ɑɓ𝐴 ⊆ 𝐽(Ѡ) ⊆  𝑉 + 𝑠𝑜𝑐(Ѡ) +
𝐽(Ѡ). That is either   ɑ𝑐𝐴 ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) +  𝐽(Ѡ)  or   ɓ𝑐𝐴 ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or  ɑɓ𝐴 ⊆
𝑉 + 𝑠𝑜𝑐(Ѡ) +  𝐽(Ѡ). Therefore by Proposition 3.4 𝑉  is  EXNPQ2AB  submodule  of Ѡ. 

4. The Relationship between the Concept of EXNPQ2AB Submodules and Other Concepts. 

The relationships between EXNPQ2AB submodules, 2-Absorbing submodules and other types of 

submodules are discussed in this section. 

Proposition 4.1 

Let Ѡ be a cyclic Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒 and 𝑉 is an 𝑒𝑠𝑠𝑒𝑛𝑡𝑖𝑎𝑙 submodule of Ѡ with 𝐽(Ѡ) ⊆ 𝑉. Then 𝑉 is 2-
Absorbing if and only if 𝑉 is EXNPQ2AB submodule of Ѡ.  

Proof. 

(⇒) By Remarks and Examples 3.2(1). 

(⇐)  Let Ѡ = Ʀ𝑤 for some 𝑤 ∈ Ѡ and assume that 𝑉 is EXNPQ2AB submodule of Ѡ. Let ɑ𝑏ℎ ∈
𝑉 for ɑ, 𝑏 ∈ Ʀ, ℎ ∈ Ѡ, then 𝑡ℎ𝑒𝑟𝑒 𝑒𝑥𝑖𝑠𝑡 an element 𝑐 ∈ Ʀ such that  ɑ𝑏ℎ = ɑ𝑏𝑐𝑤 ∈ 𝑉. Since 𝑉 is 
EXNPQ2AB submodule of Ѡ, then either ɑ𝑐𝑤 ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or 𝑏𝑐𝑤 ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) +
𝐽(Ѡ) or ɑ𝑏𝑤 ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ). That is either 𝑎𝑏 ∈ [𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ):Ʀ 𝑤] = [𝑉 +
𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ):Ʀ Ѡ] or ɑℎ ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or 𝑏ℎ ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ). Since 𝑉 is 
𝑒𝑠𝑠𝑒𝑛𝑡𝑖𝑎𝑙 submodule of Ѡ, then  𝑠𝑜𝑐(Ѡ) ⊆ 𝑉 and by hypotheses 𝐽(Ѡ) ⊆ 𝑉, we get 𝑉 +
𝑠𝑜𝑐(Ѡ) = 𝑉 and 𝑉 + 𝐽(Ѡ) = 𝑉, thus 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) = 𝑉. Hence either 𝑎𝑏 ∈ [𝑉:Ʀ Ѡ] or 
𝑏ℎ ∈ 𝑉 or ɑℎ ∈ 𝑉. Therefore 𝑉 is 2-Absorbing submodule of Ѡ. 

The Proof of the Proposition 4.2 and 4.3 is staightforward. 

Proposition 4.2 

Let Ѡ be a cyclic Ʀ-module and 𝑉 is maximal submodule of Ѡ with 𝑠𝑜𝑐(Ѡ) ⊆ 𝑉. Then 𝑉 is 2-
Absorbing if and only if 𝑉 is EXNPQ2AB submodule of Ѡ. 

 



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350 
 

Proposition 4.3 

Let Ѡ be a cyclic Ʀ-module and 𝑉 is a proper submodule of Ѡ with 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) ⊆ 𝑉. Then 𝑉 
is 2-Absorbing if and only if 𝑉 is EXNPQ2AB submodule of Ѡ. 

Proposition 4.4 

Let Ѡ be Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒 and 𝑉 is an essential submodule of Ѡ with 𝐽(Ѡ) ⊆ 𝑉. Then 𝑉 is Quasi-2-
Absorbing if and only if 𝑉 is EXNPQ-2-Absorbing.  

Proof. 

(⇒) By Remarks and Examples 3.2(2). 

(⇐) Let ɑ𝑏𝑐ℎ ∈ 𝑉 for ɑ, 𝑏, 𝑐 ∈ Ʀ and ℎ ∈ Ѡ, hence either ɑ𝑏ℎ ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or 𝑏𝑐ℎ ∈
𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or ɑ𝑐ℎ ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ). Since 𝑉 is 𝑒𝑠𝑠𝑒𝑛𝑡𝑖𝑎𝑙 submodule of Ѡ, then  
𝑠𝑜𝑐(Ѡ) ⊆ 𝑉 and by hypotheses 𝐽(Ѡ) ⊆ 𝑉, we get 𝑉 + 𝑠𝑜𝑐(Ѡ) = 𝑉 and 𝑉 + 𝐽(Ѡ) = 𝑉, thus 𝑉 +
𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) = 𝑉. Hence either ɑ𝑏ℎ ∈ 𝑉 or 𝑏𝑐ℎ ∈ 𝑉 or ɑ𝑐ℎ ∈ 𝑉. Therefor 𝑉 is Quasi-2-
Absorbing of Ѡ. 

Proposition 4.5 

Let Ѡ be Ʀ-module and 𝑉 is a 𝑝𝑟𝑜𝑝𝑒𝑟 submodule of Ѡ with 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) ⊆ 𝑉. Then 𝑉 is 
Quasi-2-Absorbing if and only if 𝑉 is EXNPQ2AB submodule of Ѡ. 

Proof. 

Direct. 

Proposition 4.6 

Let Ѡ be a cyclic Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒 with 𝑉 is a 𝑝𝑟𝑜𝑝𝑒𝑟 submodule of Ѡ and 𝑠𝑜𝑐(Ѡ) ⊆ 𝐽(Ѡ). Then 𝑉 is 
Nearly-2-Absorbing if and only if 𝑉 is EXNPQ2AB submodule of  Ѡ. 

Proof. 

(⇒) By Remarks and Examples 3.2(3). 

(⇐) Let Ѡ = Ʀ𝑤 for some 𝑤 ∈ Ѡ and assume that 𝑉 is EXNPQ2AB submodule of Ѡ. Let ɑ𝑏ℎ ∈
𝑉 for ɑ, 𝑏 ∈ Ʀ, ℎ ∈ Ѡ, then 𝑡ℎ𝑒𝑟𝑒 𝑒𝑥𝑖𝑠𝑡 an element 𝑐 ∈ Ʀ such that  ɑ𝑏ℎ = ɑ𝑏𝑐𝑤 ∈ 𝑉. Since 𝑉 is 
EXNPQ2AB submodule of Ѡ, then either ɑ𝑐𝑤 ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or 𝑏𝑐𝑤 ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) +
𝐽(Ѡ) or ɑ𝑏𝑤 ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ). That is either 𝑎𝑏 ∈ [𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ):Ʀ 𝑤] = [𝑉 +
𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ):Ʀ Ѡ] or ɑℎ ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or 𝑏ℎ ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ).  But  
𝑠𝑜𝑐(Ѡ) ⊆ 𝐽(Ѡ), thus 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) = 𝐽(Ѡ). Hence either 𝑎𝑏 ∈ [𝑉 + 𝐽(Ѡ):Ʀ Ѡ] or 𝑏ℎ ∈ 𝑉 +
𝐽(Ѡ) or ɑℎ ∈ 𝑉 + 𝐽(Ѡ). Therefore 𝑉 is Nearly-2-Absorbing submodule of Ѡ. 

Proposition 4.7 

Let Ѡ be a cyclic Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒 with 𝑉 ⊂ Ѡ and 𝑠𝑜𝑐(Ѡ) ⊆ 𝑉. Then 𝑉 is Nearly-2-Absorbing if and 
only if 𝑉 is EXNPQ2AB submodule of Ѡ.Proof. 

(⇒) Clear. 

(⇐) Let Ѡ = Ʀ𝑤 for some 𝑤 ∈ Ѡ and assume that 𝑉 is EXNPQ2AB submodule of Ѡ. Let ɑ𝑏ℎ ∈
𝑉 for ɑ, 𝑏 ∈ Ʀ, ℎ ∈ Ѡ, then 𝑡ℎ𝑒𝑟𝑒 𝑒𝑥𝑖𝑠𝑡 an element 𝑐 ∈ Ʀ such that  ɑ𝑏ℎ = ɑ𝑏𝑐𝑤 ∈ 𝑉. Since 𝑉 is 
EXNPQ2AB submodule of Ѡ, then either ɑ𝑐𝑤 ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or 𝑏𝑐𝑤 ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) +
𝐽(Ѡ) or ɑ𝑏𝑤 ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ). That is either 𝑎𝑏 ∈ [𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ):Ʀ 𝑤] = [𝑉 +
𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ):Ʀ Ѡ] or ɑℎ ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or 𝑏ℎ ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ).  Since 
𝑠𝑜𝑐(Ѡ) ⊆ 𝑉, then 𝑉 + 𝑠𝑜𝑐(Ѡ) = 𝑉, so 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) = 𝑉 + 𝐽(Ѡ). Hence either 𝑎𝑏 ∈
[𝑉 + 𝐽(Ѡ):Ʀ Ѡ] or 𝑏ℎ ∈ 𝑉 + 𝐽(Ѡ) or ɑℎ ∈ 𝑉 + 𝐽(Ѡ). Therefore 𝑉 is Nearly-2-Absorbing 
submodule of Ѡ. 

The Proof of the Proposition 4.8 and 4.9 is staightforward. 

 



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351 
 

Proposition 4.8 

Let Ѡ be a cyclic Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒 with 𝑉 is 𝑝𝑟𝑜𝑝𝑒𝑟 of Ѡ and 𝑠𝑜𝑐(Ѡ) = (0). Then 𝑉 is Nearly-2-
Absorbing if and only if 𝑉 is EXNPQ2AB submodule of Ѡ. 

Proposition 4.9 

Let Ѡ be a cyclic Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒 and 𝑉 is an 𝑒𝑠𝑠𝑒𝑛𝑡𝑖𝑎𝑙 submodule of Ѡ. Then 𝑉 is Nearly-2-
Absorbing if and only if 𝑉 is EXNPQ2AB submodule of Ѡ. 

Proposition 4.10 

Let Ѡ be Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒, 𝑠𝑜𝑐(Ѡ) ⊆ 𝐽(Ѡ) and 𝑉 ⊂ Ѡ. Then 𝑉 is Nearly Quasi-2-Absorbing if and 
only if 𝑉 is EXNPQ2AB submodule of Ѡ. 

Proof. 

(⇒)  By Remarks and Examples 3.2(4). 

(⇐) Let ɑ𝑏𝑐ℎ ∈ 𝑉 for ɑ, 𝑏, 𝑐 ∈ Ʀ, ℎ ∈ Ѡ. Since 𝑉 is EXNPQ2AB, then either ɑ𝑏ℎ ∈ 𝑉 +
𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or 𝑏𝑐ℎ ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or ɑ𝑐ℎ ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ). Since 𝑠𝑜𝑐(Ѡ) ⊆
𝐽(Ѡ), then  𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) = 𝐽(Ѡ), thus either ɑ𝑏ℎ ∈ 𝑉 + 𝐽(Ѡ) or 𝑏𝑐ℎ ∈ 𝑉 + 𝐽(Ѡ) or ɑ𝑐ℎ ∈
𝑉 + 𝐽(Ѡ). Hence 𝑉 is Nearly Quasi-2-Absorbing of Ѡ. 

Proposition 4.11 

Let Ѡ be Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒, 𝑠𝑜𝑐(Ѡ) ⊆ 𝑉 and 𝑉 ⊂ Ѡ. Then 𝑉 is Nearly Quasi-2-Absorbing if and only 
if 𝑉 is EXNPQ2AB submodule of Ѡ. 

Proof. 

Direct. 

Proposition 4.12 

Let Ѡ be a cyclic Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒 and 𝑉 is an 𝑚𝑎𝑥𝑖𝑚𝑎𝑙 submodule of Ѡ. Then 𝑉 is Pseudo-2-
Absorbing if and only if 𝑉 is EXNPQ2AB submodule of  Ѡ. 

Proof. 

(⇒)  By Remarks and Examples 3.2(5). 

(⇐) Let Ѡ = Ʀ𝑤 for some 𝑤 ∈ Ѡ and assume that 𝑉 is EXNPQ2AB submodule of Ѡ. Let ɑ𝑏ℎ ∈
𝑉 for ɑ, 𝑏 ∈ Ʀ, ℎ ∈ Ѡ, then 𝑡ℎ𝑒𝑟𝑒 𝑒𝑥𝑖𝑠𝑡 an element 𝑐 ∈ Ʀ such that ɑ𝑏ℎ = ɑ𝑏𝑐𝑤 ∈ 𝑉. Since 𝑉 is 
EXNPQ2AB submodule of Ѡ, then either ɑ𝑐𝑤 ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or 𝑏𝑐𝑤 ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) +
𝐽(Ѡ) or ɑ𝑏𝑤 ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ). That is either 𝑎𝑏 ∈ [𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ):Ʀ 𝑤] = [𝑉 +
𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ):Ʀ Ѡ] or ɑℎ ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or 𝑏ℎ ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ).  Since 𝑉 is an 
𝑚𝑎𝑥𝑖𝑚𝑎𝑙 submodule of Ѡ, then 𝐽(Ѡ) ⊆ 𝑉, hence 𝑉 + 𝐽(Ѡ) = 𝑉, so 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) = 𝑉 +
𝑠𝑜𝑐(Ѡ). Hence either 𝑎𝑏 ∈ [𝑉 + 𝑠𝑜𝑐(Ѡ):Ʀ Ѡ] or 𝑏ℎ ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) or ɑℎ ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ). 
Therefore, 𝑉 is Pseudo-2-Absorbing submodule of Ѡ. 

Proposition 4.13 

Let Ѡ be a cyclic Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒 with 𝑉 is 𝑝𝑟𝑜𝑝𝑒𝑟 of Ѡ and 𝐽(Ѡ) = (0). Then 𝑉 is Pseudo-2-
Absorbing if and only if 𝑉 is EXNPQ2AB submodule of  Ѡ. 

Proof. 

Direct. 

Proposition 4.14 

Let Ѡ be Ʀ-module and 𝑉 is a maximal submodule of Ѡ. Then, 𝑉 is Pseudo Quasi-2-Absorbing if 
and only if 𝑉 is EXNPQ2AB submodule of Ѡ. 

 



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352 
 

Proof. 

(⇒)  By Remarks and Examples 3.2(6). 

(⇐) Clear. 

Finally, we will present a Proposition that equals all the previous concepts. 

Proposition 4.15 

Let Ѡ be a multiplicatiion Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒 and 𝑉 is 𝑎 𝑝𝑟𝑜𝑝𝑒𝑟 submodule of Ѡ with 𝐽(Ѡ) = 𝑉 and 
𝑠𝑜𝑐(Ѡ) ⊆ 𝑉. Consequently, the following claims are equal: 

1. 𝑉 is a 2-Absorbing submodule of Ѡ. 

2. Quasi-2-Absorbing submodule of Ѡ. 

3. Nearly Quasi-2-Absorbing submodule of Ѡ. 

4. Nearly-2-Absorbing submodule of Ѡ. 

5. EXNPQ-2-Absorbing submodule of Ѡ. 

6. Pseudo-2-Absorbing submodule of Ѡ. 

7. Pseudo Quasi-2-Absorbing submodule of Ѡ.  

Proof. 

(𝟏 ⇔ 𝟐)  By Lemma 2.14. 

(𝟐 ⇒ 𝟑) Clear. 

(𝟑 ⇒ 𝟒) Let ɑ𝑏𝑥 ∈ 𝑉 for ɑ, 𝑏 ∈ Ʀ, 𝑥 ∈ Ѡ and assume that 𝑉 is Nearly Quasi-2-Absorbing 
submodule of Ѡ. That is ɑ𝑏(𝑥) ⊆ 𝑉, since Ѡ is a 𝑚ultiplication Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒, then 𝑡ℎ𝑒𝑟𝑒 𝑒𝑥𝑖𝑠𝑡 an 
ideal 𝐼 of Ʀ such that (𝑥) = 𝐼Ѡ, hence ɑ𝑏𝐼Ѡ ⊆ 𝑉. Since 𝑉 is Nearly Quasi-2-Absorbing 
submodule of Ѡ, then either ɑ𝐼Ѡ ⊆ 𝑉 + 𝐽(Ѡ) or 𝑏𝐼Ѡ ⊆ 𝑉 + 𝐽(Ѡ) or ɑ𝑏Ѡ ⊆ 𝑉 + 𝐽(Ѡ). That is 
either 𝑎𝑏 ∈ [𝑉 + 𝐽(Ѡ):Ʀ Ѡ] or ɑ𝑥 ∈ 𝑉 + 𝐽(Ѡ) or 𝑏𝑥 ∈ 𝑉 + 𝐽(Ѡ). Therefore 𝑉 is Nearly-2-
Absorbing submodule of Ѡ. 

(𝟒 ⇒ 𝟓) By Remarks and Examples 3.2(3). 

(𝟓 ⇒ 𝟔) Let ɑ𝑏𝑥 ∈ 𝑉 for ɑ, 𝑏 ∈ Ʀ, 𝑥 ∈ Ѡ and assume that 𝑉 is EXNPQ2AB submodule of Ѡ. 
That is ɑ𝑏(𝑥) ⊆ 𝑉, since Ѡ is a 𝑚ultiplicatiIon Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒, then 𝑡ℎ𝑒𝑟𝑒 𝑒𝑥𝑖𝑠𝑡 an ideal 𝐼 of Ʀ such 
that (𝑥) = 𝐼Ѡ, hence ɑ𝑏𝐼Ѡ ⊆ 𝑉. Since 𝑉 is EXNPQ2AB submodule of Ѡ, then either ɑ𝐼Ѡ ⊆
𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or 𝑏𝐼Ѡ ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or ɑ𝑏Ѡ ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ). That is 
either 𝑎𝑏 ∈ [𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ):Ʀ Ѡ] or ɑ𝑥 ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) or 𝑏𝑥 ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) +
𝐽(Ѡ). But 𝐽(Ѡ) ⊆ 𝑉, then 𝑉 + 𝐽(Ѡ) = 𝑉 and 𝑉 + 𝑠𝑜𝑐(Ѡ) + 𝐽(Ѡ) = 𝑉 + 𝑠𝑜𝑐(Ѡ). Thus either 
𝑎𝑏 ∈ [𝑉 + 𝑠𝑜𝑐(Ѡ) ∶Ʀ Ѡ] or 𝑏𝑥 ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) or ɑ𝑥 ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ). Therefore 𝑉 is Pseudo-2-
Absorbing submodule of Ѡ. 

(𝟔 ⇒ 𝟕) Let ɑɓ𝑐ӽ ∈ 𝑉, for ɑ,ɓ, 𝑐 ∈ Ʀ, ӽ ∈ Ѡ. That is ɑɓ(𝑐ӽ) ∈ 𝑉. But 𝑉 is Pseudo-2-Absorbing 
submodule of  Ѡ, then either ɑ(𝑐ӽ) ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) or ɓ(𝑐ӽ) ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ)  or ɑɓѠ ⊆ 𝑉 +
𝑠𝑜𝑐(Ѡ). Thus either ɑ𝑐ӽ ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) or ɓ𝑐ӽ ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) or ɑɓӽ ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) for all ӽ ∈
Ѡ. Hence 𝑉 is Pseudo Quasi-2-Absorbing submodule of  Ѡ. 

(𝟕 ⇒ 𝟏) Let ɑ𝑏𝑥 ∈ 𝑉 for ɑ, 𝑏 ∈ Ʀ, 𝑥 ∈ Ѡ and assume that 𝑉 is Pseudo Quasi-2-Absorbing 
submodule of Ѡ. That is ɑ𝑏(𝑥) ⊆ 𝑉, since Ѡ is a 𝑚ultiplication Ʀ-𝑚𝑜𝑑𝑢𝑙𝑒, then tℎ𝑒𝑟𝑒 𝑒𝑥𝑖𝑠𝑡 an 
ideal 𝐼 of Ʀ such that (𝑥) = 𝐼Ѡ, hence ɑ𝑏𝐼Ѡ ⊆ 𝑉. Since 𝑉 is Pseudo Quasi-2-Absorbing 
submodule of Ѡ, then either ɑ𝐼Ѡ ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ)or 𝑏𝐼Ѡ ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ)or ɑ𝑏Ѡ ⊆ 𝑉 + 𝑠𝑜𝑐(Ѡ). 
That is either 𝑎𝑏 ∈ [𝑉 + 𝑠𝑜𝑐(Ѡ):Ʀ Ѡ] or ɑ𝑥 ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ) or 𝑏𝑥 ∈ 𝑉 + 𝑠𝑜𝑐(Ѡ). But 𝑠𝑜𝑐(Ѡ) ⊆
𝑉, hence 𝑉 + 𝑠𝑜𝑐(Ѡ) = 𝑉. Thus either 𝑎𝑏 ∈ [𝑉:Ʀ Ѡ] or 𝑏𝑥 ∈ 𝑉 or ɑ𝑥 ∈ 𝑉. Therefore, 𝑉 is 2-
Absorbing submodule of Ѡ. 



IHJPAS. 36(2)2023 

353 
 

5. Conclusion 

The term EXNPQ2AB submodules is a novel generalization of (2-Absorbing, Quasi-2-Absorbing, 

Nearly-2-Absorbing, Nearly Quasi-2-Absorbing, Pseudo-2-Absorbing and Pseudo Quasi-2-

Absorbing) submodules that we introduce in this article. Using examples, we also discuss the 

opposite generalization introduced in several different characterizations. There are given 

connections between this generalization and other types of modules. 

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