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

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Weakly Nearly Prime Submodules  
  

 Ali Sabah Sadiaq                                 Haibat K. Mohammadali 

Department of Mathematics, College of Computer Science and Math. , University of  Tikrit, Tikrit, Iraq. 

H.mohammadali@tu.edu.iq alisabahsadickali@gmail.com 

 

 

 

 Abstract 

        In this article, unless otherwise established, all rings are commutative with identity and all 

modules are unitary left R-module. We offer this concept of WN-prime as new generalization of 

weakly prime submodules. Some basic properties of weakly nearly prime submodules are given. 

Many characterizations, examples of this concept are stablished. 

 

 Keywords: Weakly prime submodules, weakly nearly prime submodules, multiplication modules, 

finitely generated modules, Jacobson of a modules. 

 

1.Introduction 

        The concept of weakly prime submodule   was first Introduced  and studied by Behoodi and 

Koohi in [1] as a generalization of weakly prime submodule , where a proper submodule 𝐻 of an 

R-module   𝑈  is weakly prime submodule, if whenever 0 ≠ 𝑟𝑢 ∈ 𝐻, for 𝑟 ∈ 𝑅,𝑢 ∈  𝑈, implies 

that  either 𝑢 ∈ 𝐻 or 𝑟 𝑈 ⊆ 𝐻. Recently, weakly prime submodules have been studied by many 

authors such as [2-5]. Many generalizations of weakly prime submodule are introduced such as 

weakly primary submodules, weakly quasi- prime submodules and weakly semi- prime 

submodules see [6- 8]. In 2018 the concepts WE-prime submodules and WE-semi- prime 

submodules as a strange from of weakly prime submodules are given; see [9]. In this article, we 

introduce a new generalization of weakly prime submodule called WN-prime submodule , where 

a proper submodule 𝐻 of an 𝑅-module  𝑈 is called WN-prime of  𝑈 if whenever  0 ≠ 𝑟𝑢 ∈ 𝐻, for  

𝑟 ∈ 𝑅, 𝑢 ∈  𝑈, implies that either 𝑢 ∈ 𝐻 + 𝙹( 𝑈) or 𝑟 𝑈 ⊆ 𝐻 + 𝙹( 𝑈), where 𝙹( 𝑈) is the 

Jacobson radical of  𝑈. An R-module   𝑈 is multiplication if each submodule 𝐻 of  𝑈 from  

𝐻 = 𝐼 𝑈 for some ideal 𝐼  of 𝑅 , that is  𝐻 = [𝐻:𝑅  𝑈] 𝑈 [10]. Several characterizations, examples 

and basic properties of WN-prime submodules were given in this research. 

 

Ibn Al Haitham Journal for Pure and Applied Science 

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

 

Doi: 10.30526/34.1.2556 

Article history: Received 16,January,2020, Accepted12,February,2020, Published in January 2021 

 

mailto:H.mohammadali@tu.edu.iq
mailto:H.mohammadali@tu.edu.iq
mailto:alisabahsadickali@gmail.com


  

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

 

 

2.  Basic Properties of Weakly Nearly Prime Submodules  

       In this stage, we offer the definition of weakly nearly prime submodule and establish some of 

its basic properties and characterizations.  

 

Definition (2.1) 

     A proper submodule 𝐻 of 𝑅-module  𝑈 is said to be weakly nearly  prime submodule of  𝑈 

(for short WN-prime  submodule), if whenever 0 ≠ 𝑎𝑢 ∈ 𝐻, where 𝑎 ∈ 𝑅, 𝑢 ∈  𝑈, implies that 

either 𝑢 ∈ 𝐻 + 𝙹( 𝑈) or 𝑟 𝑈 ⊆ 𝐻 + 𝙹( 𝑈).An ideal 𝐴 of ring 𝑅 is WN-prime ideal of 𝑅 if and 

only if  𝐴 is a WN-prime submodule of an 𝑅-module 𝚁. 

 For example : consider the Z-module  𝑍24  and the submodule 𝐻 = 〈8̅〉 of 𝑍24 which is a WN-

prime submodule of 𝑍24 since 𝙹(𝑍24) = 〈2̅〉 ∩ 〈3̅〉 = 〈6̅〉. Thus if 0 ≠ 𝑟𝑚 ∈ 𝐻 with 𝑟 ∈ 𝑍, 

𝑚 ∈ 𝑍24 , implies that either 𝑚 ∈ 𝐻 + 𝙹(𝑍24) = 〈8̅〉 + 〈6̅〉 = 〈2̅〉  or 𝑟 ∈ [𝐻 + 𝙹(𝑍24):𝑍24] =

[〈2̅〉:𝑍24] = 2𝑍 . 

 

Remark (2.2) 

 1. It is clear that every weakly  prime submodule of an R-module  𝘜 is WN-prime , but not 

conversely.  

 For example the submodule 𝑁 = 𝑍 of the Z-module 𝑄 is not weakly prime,  but 𝑁 is WN-prime 

since 𝙹(𝑄) = 𝑄 and for each  𝑎 ∈ 𝑍, 𝑢 ∈ 𝑄 with 0 ≠ 𝑎𝑢 ∈ 𝑁 , implies that either 𝑢 ∈ 𝑁 + 𝙹(𝑄) 

or 𝑎𝑄 ⊆ 𝑍 + 𝙹(𝑄) = 𝑄 . 

 

2. It is clear that every  prime submodule of an R-module   𝑈 is WN-prime, but not conversely .  

For example : consider that the Z-module  𝑍12 , and the submodule 𝐻 = 〈4̅〉 of 𝑍12 is not  prime , 

but 𝐻 = 〈4̅〉 is WN-prime submodule of 𝑍12 since 𝙹(𝑍12) = 〈2̅〉 ∩ 〈3̅〉 = 〈6̅〉. Thus if 0 ≠ 𝑟𝑢 ∈ 𝐻 

with 𝑟 ∈ 𝑍, 𝑢 ∈ 𝑍12 , implies that either 𝑢 ∈ 𝐻 + 𝙹(𝑍12) = 〈4̅〉 + 〈6̅〉 = 〈2̅〉  or 𝑟 ∈ [𝐻 +

𝙹(𝑍12):𝑍12] = [〈2̅〉:𝑍12] = 2𝑍 . 

 

3. If 𝐻 is proper submodule of an R-module   𝑈 with 𝙹( 𝑈) ⊆ 𝐻.  Then 𝐻  is a WN-prime if and 

only if 𝐻 is weakly prime submodule . 

 

4.If  𝑈 is a semi-simple R-module  and 𝐻 is a proper submodule of  𝑈,then 𝐻 is a weakly  prime if 

and only if  𝐻  is WN-prime submodule of  𝑈. 

Proof  

      It is well-known if  𝑈 is a semi-simple, then 𝙹( 𝑈) = (0) . [14, Theo. (9.2.1) (a)]. So the proof 

follows direct. 

 

The following propositions give characterizations of WN-prime submodules.  

Proposition (2.3)  

       Let  𝑈 be an 𝑅-module, 𝐻 be a submodule of   𝑈, then 𝐻 is a WN-prime submodule of  𝑈 if 

and only if for every submodule 𝐿 of  𝑈 and 𝑟 ∈ 𝑅 with  0 ≠ 〈𝑟〉𝐿 ⊆ 𝐻, implies that either 

𝐿 ⊆ 𝐻 + 𝙹( 𝑈) or 〈𝑟〉 𝑈 ⊆ 𝐻 + 𝙹( 𝑈) . 

 



  

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Proof 

     (
        
⇒ ) Suppose that 0 ≠ 〈𝑟〉𝐿 ⊆ 𝐻, for 𝑟 ∈ 𝑅 , and 𝐿 is a submodule of  𝑈, with 𝐿 ⊈ 𝐻 + 𝙹( 𝑈), 

then 𝑙 ∉ 𝐻 + 𝙹( 𝑈)  for some non-zero element 𝑙 ∈ 𝐿. Now 0 ≠ 𝑟𝑙 ∈ 𝐻, then since 𝐻 is WN-

prime submodule of  𝑈, and 𝑙 ∉ 𝐻 + 𝙹( 𝑈), then we have  𝑟 ∈ [𝐻 + 𝙹( 𝑈): 𝑈] ,  it follows that   

〈𝑟〉 ⊆ [𝐻 + 𝙹( 𝑈): 𝑈]. That is 〈𝑟〉 𝑈 ⊆ 𝐻 + 𝙹( 𝑈) 

 

    (
         
⇐  ) Let  0 ≠ 𝑟𝑢 ∈ 𝐻, for 𝑟 ∈ 𝑅, 𝑢 ∈  𝑈, it follows that   0 ≠ 〈𝑟〉〈𝑢〉 ⊆ 𝐻, so by hypothesis 

either 〈𝑢〉 ⊆ 𝐻 + 𝙹( 𝑈) or 〈𝑟〉 𝑈 ⊆ 𝐻 + 𝙹( 𝑈). That is either 𝑢 ∈ 𝐻 + 𝙹( 𝑈) or 𝑟 𝑈 ⊆ 𝐻 +

𝙹( 𝑈). Hence 𝐻  is a WN-prime submodule of  𝑈. 

 

 As direct result of Proposition (2.3) we get the following corollary. 

 

Corollary (2.4) 

     A proper submodule 𝐻 of an R-module   Ų is WN-prime if and only if  for every submodule 𝐾 

of  𝑈 and every  𝑟 ∈ 𝑅 such that   0 ≠ 𝑟𝐾 ⊆ 𝐻, implies that either 𝐾 ⊆ 𝐻 + 𝙹( 𝑈) or 𝑟 ∈ [𝐻 +

𝙹( 𝑈) ∶  𝑈] . 

 

Proposition (2.5) 

       Let 𝐻 be proper submodule of R-module  𝑈, then 𝐻 is WN-prime submodule of  𝑈 if and, 

only if [𝐻:𝑅 𝑥] ⊆ [𝐻 + 𝙹( 𝑈):𝑅  𝑈] ∪ [0:𝑅 𝑥] for all 𝑥 ∈  𝑈 and 𝑥 ∉ 𝐻 + 𝙹( 𝑈). 

Proof 

     (
        
⇒ ) Let 𝑟 ∈ [𝐻:𝑅 𝑥]  and 𝑥 ∉ 𝐻 + 𝙹( 𝑈), then 𝑟𝑥 ∈ 𝐻 .If 𝑟𝑥 ≠ 0, and 𝐻 is a WN-prime 

submodule of  𝑈  and  𝑥 ∉ 𝐻 + 𝙹( 𝑈) , hence 𝑟 ∈ [𝐻 + 𝙹( 𝑈):𝑅  𝑈]. If 𝑟𝑥 = 0, then 𝑟 ∈

[0:𝑅 𝑥].Thus 𝑟 ∈ [𝐻 + 𝙹( 𝑈):𝑅  𝑈] ∪ [0:𝑅 𝑥].Hence [𝐻:𝑅 𝑥] ⊆ [𝐻 + 𝙹( 𝑈):𝑅  𝑈] ∪ [0:𝑅 𝑥]. 

      (
         
⇐  ) Let 0 ≠ 𝑟𝑥 ∈ 𝐻 for 𝑟 ∈ 𝑅 ,𝑢 ∈  𝑈, with 𝑥 ∉ 𝐻 + 𝙹( 𝑈), then 𝑟 ∈ [𝐻:𝑅 𝑥], by hypothesis 

𝑟 ∈ [𝐻 + 𝙹( 𝑈):𝑅  𝑈] ∪ [0:𝑅 𝑥], but 𝑟𝑥 ≠ 0. Thus, 𝑟 ∈ [𝐻 + 𝙹( 𝑈):𝑅  𝑈] and hence 𝐻 is a WN-

prime submodule of  𝑈 . 

 

Proposition (2.6) 

        Let H be a proper submodule of  an R-module   𝑈  with [𝐻 + 𝙹( 𝑈):𝑅  𝑈] is a maximal ideal 

of  𝑅, then 𝐻 is a WN-prime submodule of  𝑈. 

Proof 

       Suppose that 0 ≠ 𝑟𝑢 ∈ 𝐻 , with 𝑟 ∈ 𝑅 ,𝑢 ∈  𝑈 and 𝑟 𝑈 ⊈ 𝐻 + 𝙹( 𝑈). That is, 𝑟 ∉

 [𝐻 + 𝙹( 𝑈): 𝑈],but   [𝐻 + 𝙹( 𝑈): 𝑈] is maximal, then by [11,Th. 5.1] 𝑅 = 〈𝑟〉 + [𝐻 +

𝙹( 𝑈):𝑅  𝑈]. It follows that 1 = 𝑎𝑟 + 𝑏, for some 𝑎 ∈ 𝑅 , 𝑏 ∈ [𝐻 + 𝙹( 𝑈):𝑅  𝑈]. Hence, 𝑢 =

𝑎𝑟𝑢 + 𝑏𝑢 ∈ 𝐻 + 𝙹( 𝑈). Hence, 𝐻 is a WN-prime submodule of  𝑈. 

 

Proposition (2.7) 

        Let 𝐻 be a proper submodule of an R-module  𝘜 with  [𝐿:𝑅 𝘜] ⊈ [𝐻 + 𝙹(𝘜):𝑅 𝘜] and 

𝐻 + 𝙹(𝘜) is a proper submodule of 𝐿 for each submodule 𝐿 of 𝘜 .If  [𝐻 + 𝙹(𝘜):𝑅 𝘜] is a  prime 

ideal of 𝑅,  then 𝐻 is a WN-prime submodule of 𝘜. 



  

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

 

 

 

Proof  

       Assume that 0 ≠ 𝑟𝑢 ∈ 𝐻 , for 𝑟 ∈ 𝑅 ,𝑢 ∈ 𝘜 and 𝑢 ∉ 𝐻 + 𝙹(𝘜).We have 𝐻 + 𝙹(𝘜) ⊈ 𝐻 +

𝙹(𝘜) + 〈𝑢〉 , put 𝐿 = 𝐻 + 𝙹(𝘜) + 〈𝑢〉 = 𝐿, then [𝐿:𝑅 𝘜] ⊈ [𝐻 + 𝙹(𝘜):𝑅  𝑈]. That is there exist 

𝑎 ∈ [𝐿:𝑅 𝘜] and 𝑎 ∉ [𝐻 + 𝙹(𝘜):𝑅 𝘜]. It follows that 𝑎𝘜 ⊆ 𝐿 but 𝑎𝘜 ⊈ 𝐻 + 𝙹𝘜. 𝑎𝘜 ⊆ 𝐿, implies 

that 𝑟𝑎𝘜 ⊆ 𝑟𝐿 = 𝑟(𝐻 + 𝙹(𝘜) + 〈𝑢〉) ⊆ 𝐻 + 𝙹(𝘜), that is 𝑟𝑎 ∈ [𝐻 + 𝙹(𝘜):𝘜]. But  [𝐻 +

𝙹(𝘜):𝑅 𝘜] is a  prime ideal of 𝑅 and 𝑎 ∉ [𝐻 + 𝙹(𝘜):𝑅 𝘜] then 𝑟 ∈ [𝐻 + 𝙹(𝘜) ∶ 𝘜]. Thus 𝐻 is a 

WN-prime submodule of 𝘜. 

 

        It is well-known that if  𝑈 is a multiplication R-module  and 𝐻 is a proper submodule of  𝑈, 

then [𝐿:𝑅  𝑈] ⊈ [𝐻 :𝑅  𝑈] for each submodule 𝐿 of  𝑈 with 𝐻 ⊈ 𝐿 [12, Rem. (2.15)]. 

Corollary (2.8) 

        Let 𝐻 be a proper submodule of a multiplication R-module   𝑈,then 𝐻 is a WN-prime 

submodule of  𝑈, if  [𝐻 + 𝙹( 𝑈):𝑅  𝑈] is a  prime ideal of 𝑅 and 𝐻 + 𝙹( 𝑈) is a proper submodule 

of 𝐿 for each submodule 𝐿 of  𝑈. 

 

If 𝐻 is a submodule of an 𝑅-module  𝑈,  then 𝐻(𝑆) = {𝑢 ∈  𝑈:∃𝑡 ∈ 𝑆 such that 𝑡𝑢 ∈ 𝐻} [13]. 

 

Proposition (2.9)  

         Let 𝐻 be a proper submodule of an R-module   𝘜, with [𝐻 + 𝙹( 𝘜):𝑅  𝘜] is a prime ideal of 

𝑅, then 𝐻 is  WN-prime if and only if 𝐻(𝑆) ⊆ 𝐻 + 𝙹( 𝘜) for each  multiplicatively closed subset 

𝑆  of 𝑅 with 𝑆 ∩ [𝐻 + 𝙹( 𝘜):𝑅  𝘜] = 𝜑. 

Proof  

     (
        
⇒ ) Suppose that 𝐻 is a  WN-prime submodule of  𝘜 with  𝑆 ∩ [𝐻 + 𝙹( 𝘜):𝑅  𝘜] = 𝜑. Let 

𝑢 ∈ 𝐻(𝑆), then ∃𝑟 ∈ 𝑆 such that 𝑟𝑢 ∈ 𝐻, implies that 𝑟 ∈ [𝐻:𝑅 𝑢] ⊆ [𝐻 + 𝙹( 𝘜):𝑅  𝘜] ∪ [0:𝑅 𝑢] 

by Proposition (2.5) .It follows that 0 ≠ 𝑟𝑢 ∈ 𝐻 (since 𝐻 is a WN-prime ), implies that either 

𝑢 ∈ 𝐻 + 𝙹( 𝑈) or 𝑟 ∈ [𝐻 + 𝙹( 𝑈):𝑅  𝘜]. If 𝑟 ∈ [𝐻 + 𝙹( 𝘜):𝑅  𝘜], implies that 𝑟 ∈  𝑆 ∩

[𝐻 + 𝐽( 𝘜):𝑅 𝘜] = 𝜑 which is a contradiction. Thus 𝑢 ∈ 𝐻 + 𝙹( 𝘜) and hence 𝐻(𝑆) ⊆ 𝐻 + 𝙹( 𝘜). 

     (
         
⇐  )  Suppose that 0 ≠ 𝑟𝑢 ∈ 𝐻 where 𝑟 ∈ 𝑅 ,𝑢 ∈  𝘜 such that 𝑢 ∉ 𝐻 + 𝙹( 𝘜) and 𝑟 ∉

[𝐻 + 𝙹( 𝘜):𝑅  𝘜]. Since 𝑟 ∈ 𝑆, then 𝑆 = {1,𝑟,𝑟
2,𝑟3,…} is multiplicatively closed subset of 𝑅 and 

𝑆 ∩ [𝐻 + 𝙹( 𝘜):𝑅  𝘜] = 𝜑 (since[𝐻 + 𝙹( 𝘜):𝑅  𝘜] is  prime ideal of 𝑅 ). But  𝑢 ∉ 𝐻 + 𝙹( 𝘜) 

implies that 𝑢 ∉ 𝐻(𝑆) and then 0 ≠ 𝑟𝑢 ∉ 𝐻 which is a contradiction. Thus 𝑢 ∈ 𝐻 + 𝙹( 𝘜) or 

𝑟 ∈ [𝐻 + 𝙹( 𝘜):𝑅  𝘜]. That is, 𝐻 is a  WN-prime submodule of  𝘜. 

 

The following corollary a direct consequence of Proposition (2.9). 

Corollary (2.10) 

       Let  𝑈 be an 𝑅-module, 𝐻 be a proper submodule of  𝑈, with [𝐻 + 𝙹( 𝑈):𝑅  𝑈] is  prime ideal 

in 𝑅, then 𝐻 is WN-prime if and only  if 𝐻(𝑅 − ([𝐻 + 𝙹( 𝑈):𝑅  𝑈]) ⊆ 𝐻 + 𝙹( 𝑈). 

 

Proposition (2.11)  

       Let  𝑈 be an 𝑅-module, and 𝐴 be a maximal ideal of  𝑅, with 𝐴 𝑈 + 𝙹( 𝑈) ≠  𝑈. Then 𝐴 𝑈 is 

a WN-prime submodule of   𝑈. 

Proof: 



  

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

 

       Since 𝐴 𝑈 ⊆  𝐴 𝑈 + 𝙹( 𝑈), then 𝐴 ⊆[A U+J( U) :𝑅U] .That is, there exists 𝑟 ∈ [𝐴 𝑈 +

𝙹( 𝑈): 𝑈] and 𝑟 ∉ 𝐴. But 𝐴 is a maximal ideal of  𝑅, then 𝑅 = 𝐴 + 〈𝑟〉, then  1 = 𝑎 + 𝑠𝑟 for some 

𝑠 ∈ 𝑅, it follows that 𝑢 = 𝑎𝑢 + 𝑠𝑟𝑢 for each 𝑢 ∈  𝑈. Thus 𝑢 ∈ 𝐴 𝑈 + 𝙹( 𝑈) for each 𝑢 ∈  𝑈, so 

𝐴 𝑈 + 𝙹( 𝑈) =  𝑈 which is a contradiction. Hence, 𝑟 ∈ 𝐴 and it follows that [𝐴 𝑈 + 𝙹( 𝑈): 𝑈] ⊆

𝐴.Thus [𝐴 𝑈 + 𝙹( 𝑈): 𝑈] = 𝐴. That is, [𝐴 𝑈 + 𝙹( 𝑈): 𝑈] is a maximal ideal of  𝑅, hence by 

Proposition( 2.6 ), 𝐴 𝑈 is  a WN-prime submodule of   𝑈. 

 

Proposition (2.12) 

       Let 𝐻 be a proper submodule of an R-module   𝑈 with  [𝐻 + 𝙹( 𝑈):𝑅  𝑈] = [𝐻 + 𝙹( 𝑈):𝑅 𝐾] 

for each submodule 𝐾 of  𝑈 such that  𝐻 + 𝙹( 𝑈) is a proper submodule of 𝐿, then 𝐻 is a WN-

prime submodule of  𝑈. 

Proof 

      Suppose that 0 ≠ 𝑟𝑢 ∈ 𝐻 for each 𝑟 ∈ 𝑅 ,u ∈ U with 𝑢 ∉ 𝐻 + 𝙹( 𝑈). Assume that 𝐾 = 𝐻 +

𝙹( 𝑈) + 〈𝑢〉, it is clear that 𝐻 + 𝙹( 𝑈) ⊆ 𝐾, then  𝑢 ∈ 𝐾 and so 𝑟 ∈ [𝐻:𝑅 𝐾]. Since 𝐻 ⊆ 𝐻 +

𝙹( 𝑈), then [𝐻:𝑅 𝐾] = [𝐻 + 𝙹( 𝑈):𝑅 𝐾] = [𝐻 + 𝙹( 𝑈):𝑅  𝑈] by hypothesis. Thus 𝑟 ∈

[𝐻 + 𝙹( 𝑈):𝑅  𝑈], it follow that  𝐻 is  a WN-prime submodule of  𝑈. 

 

Recall that submodule 𝐻 of an R-module   𝑈 is to said to be  small, if  for any submodule 𝐾 of  𝑈 

with  𝑈 = 𝐻 + 𝐾 then 𝐾 =  𝑈 [14]. 

Proposition (2.13) 

        Let 𝐻 be a small proper submodule of an R-module  𝘜 and 𝙹(𝘜) is a weakly  prime 

submodule of 𝘜, then 𝐻 is a WN-prime submodule of 𝘜. 

Proof 

      Suppose that 0 ≠ 𝑟𝑢 ∈ 𝐻, where 𝑟 ∈ 𝑅 ,𝑢 ∈ 𝘜. Since 𝐻 is a small submodule of 𝘜, then 

0 ≠ 𝑟𝑢 ∈ 𝐻 ⊆ 𝙹(𝘜). It follows that  0 ≠ 𝑟𝑢 ∈ 𝙹(𝘜), but 𝙹(𝘜) is a weakly  prime submodule of 

 𝑈, implies that  either 𝑢 ∈ 𝙹(𝘜) ⊆ 𝐻 + 𝙹(𝘜) or 𝑟𝘜 ⊆ 𝙹(𝘜) ⊆ 𝐻 + 𝙹(𝘜). Hence 𝐻 is a WN-

prime submodule of 𝘜. 

  

Remark (2.14) 

        If 𝐻 and 𝐿 are two  submodules  of R-module   𝑈 with 𝐻 is  contained in 𝐿, 𝐿 is a WN-prime 

submodule of   𝑈. Then 𝐻 not necessary  to be WN-prime submodule of  𝑈. The following 

example explains that. Consider the Z-module 𝑍24 and the submodule 𝐻 = {0̅ ,12̅̅̅̅  }, 𝐿 =

{0̅, 2̅ , 4̅, 6̅, 8̅,10̅̅̅̅ ,12̅̅̅̅ ,14̅̅̅̅ ,16̅̅̅̅ ,18̅̅̅̅ ,20̅̅̅̅ ,22̅̅̅̅ } we have 𝐿 is a WN-prime (since 𝐿 is a weakly  prime ) 

submodule of the Z-module  𝑍24, but 𝐻 is not WN-prime because if 3 ∈ 𝑍,4̅ ∈ 𝑍24 such that 

0̅ ≠ 3 4̅ ∈ 𝐻, but 4̅ ∉ 𝐻 + 𝙹(𝑍24) = {0̅, 6̅,12̅̅̅̅ ,18̅̅̅̅ } and 3 ∉ [ 𝐻 + 𝙹(𝑍24) ∶ 𝑍24] = 6𝑍. 

 

Proposition (2.15) 

        Let  𝑈 be an R-module , and 𝐻,𝐿 are submodules of  𝑈 with 𝐻 contained in  𝐿, and 𝙹( 𝑈) ⊆

𝙹(𝐿). If 𝐻 is WN-prime submodule of  𝑈, then 𝐻 is WN-prime submodule of 𝐿. 

 

Proof 

      Assume that 0 ≠ 𝑟𝑥 ∈ 𝐻 with 𝑟 ∈ 𝑅,𝑥 ∈ 𝐿. Since  𝐿 is a WN-prime submodule of  𝑈, 

then 𝑥 ∈ 𝐻 + 𝙹( 𝑈) or 𝑟 ∈ [𝐻 + 𝙹( 𝑈):𝑅  𝑈]. But 𝙹( 𝑈) ⊆ 𝙹(𝐿) so   𝑥 ∈ 𝐻 + 𝙹(𝐿) or 𝑟 ∈

[𝐻 + 𝙹(𝐿):𝑅  𝑈] ⊆ [𝐻 + 𝙹(𝐿):𝑅 𝐿]. Hence 𝐻 is a WN-prime submodule of 𝐿. 



  

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Remark (2.16) 

         The resudule of WN-prime submodule of an R-module   𝑈 need not to be WN-prime ideal of 

𝑅. The following example shows that: 

Let  𝑈 = 𝑍12 , 𝑅 = 𝑍  and 𝐻 = {0̅, 4̅, 8̅}, 𝐻 is a WN-prime submodule of 𝑍12 by Remark(2.2)(2). 

But [𝐻:𝑍 𝑍12] = 4𝑍 is not WN-prime ideal of 𝑅 because 0 ≠ 2 2 ∈ 4𝑍,2 ∈ 𝑍 but 2 ∉ 4𝑍 +

𝙹(𝑍) = 4𝑍 and 2 ∉ [4𝑍 + 𝙹(𝑍):𝑍 𝑍] = 4𝑍. 

 

        The following propositions show that the resudule of a WN-prime submodule is a WN-prime 

ideal in the class of multiplication R-module over a good ring, Artinian ring respectively. 

 

Remember  that A ring 𝑅 is called good if 𝙹(𝘜) = 𝙹(𝑅).𝘜 where 𝘜 is an R-module  [14]. 

 

Proposition (2.17) 

         Let 𝘜 be a multiplication module  over a good ring 𝑅 , and  𝐻 is a WN-prime submodule of 

𝘜 then [𝐻:𝑅 𝘜] is  a WN-prime ideal of 𝑅. 

 

Proof 

      suppose that 0 ≠ 𝑟𝑠 ∈ [𝐻:𝑅 𝘜] where 𝑟,𝑠 ∈ 𝑅, implies that  0 ≠ 𝑟(𝑠𝘜) ⊆ 𝐻. But 𝐻 is a WN-

prime submodule of 𝘜, then by Corollary (2.4) either 𝑠𝘜 ⊆ 𝐻 + 𝙹(𝘜) or 𝑟𝘜 ⊆ 𝐻 + 𝙹(𝘜). For 𝘜  

a multiplication module  over good ring, then 𝙹(𝘜) = 𝙹(𝑅).𝘜 and 𝐻 = [𝐻:𝑅 𝘜].𝘜. Thus either 

𝑠𝘜 ⊆ [𝐻:𝑅 𝘜].𝘜 + 𝙹(𝑅).𝘜 or 𝑟𝘜 ⊆ [𝐻:𝑅 𝘜] 𝑈 + 𝙹(𝑅)𝘜. Hence either 𝑠 ∈ [𝐻:𝑅 𝘜] + 𝙹(𝑅) or 

𝑟 ∈ [𝐻:𝑅 𝘜] + 𝙹(𝑅) = [[𝐻:𝑅 𝘜] + 𝙹(𝑅):𝑅 𝘜]. Therefore [𝐻:𝑅 𝘜] is a  WN-prime ideal of 𝑅. 

 

     It is well known if  𝑈 is a module  over Artinian ring 𝑅 then 𝙹(𝘜) = 𝙹(𝑅)𝘜. [14, Co. 

9.3.10(c)]. 

 

Proposition (2.18)  

       Let 𝘜 is a multiplication module  over Artinian ring 𝑅, and  𝐻 is  a WN-prime submodule of 

 𝑈 then  [𝐻:𝑅  𝑈] is  a WN-prime ideal of 𝑅. 

Proof  

       Let 0 ≠ 𝑟𝐼 ∈ [𝐻:𝑅 𝘜] where 𝑟 ∈ 𝑅 and 𝐼 is an ideal of 𝑅,  then  0 ≠ 𝑟𝐼 ⊆ 𝐻. Since 𝐻 is a 

WN-prime submodule of  𝑈, then by Corollary (2.4) either 𝐼𝘜 ⊆ 𝐻 + 𝙹(𝘜) or 𝑟𝘜 ⊆ 𝐻 +

𝙹(𝘜).But  𝑈 is   a multiplication module  over good ring 𝑅, then 𝙹(𝘜) = 𝙹(𝑅)𝘜 and 𝐻 =

[𝐻:𝑅 𝘜]𝘜. It follows that either 𝐼𝘜 ⊆ [𝐻:𝑅 𝘜]𝘜 + 𝙹(𝑅)𝘜 or 𝑟𝘜 ⊆ [𝐻:𝑅 𝘜]𝘜 + 𝙹(𝑅).𝘜. Hence 

either 𝐼 ⊆ [𝐻:𝑅 𝘜] + 𝙹(𝑅) or 𝑟 ∈ [𝐻:𝑅 𝘜] + 𝙹(𝑅) = [[𝐻:𝑅 𝘜] + 𝙹(𝑅):𝑅 𝘜].Therefore [𝐻:𝑅  𝑈] is 

a  WN-prime ideal of 𝑅. 

 

       It is well known that if 𝘜 is a projective R-module  then 𝙹(𝘜) = 𝙹(𝑅).𝘜 [14, Th. 9.2.1(g)]. 

Proposition (2.19) 

           Let 𝘜 be  a projective multiplication  R-module , and  𝐻 is a WN-prime submodule of 𝘜 

then  [𝐻:𝑅 𝘜] is a  WN-prime ideal of 𝑅. 

Proof  



  

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      Follows in the same way of Proposition (2.17) and Proposition (2.18). 

 

      It is well known if  𝑈 is a multiplication finitely generated R-module , and 𝐴,𝐵 are ideals of 𝑅, 

then 𝐴 𝑈 ⊆ 𝐵 𝑈 if and only if 𝐴 ⊆ 𝐵 + 𝑎𝑛𝑛( 𝑈) [15, Cor. of th. 9]. 

 

Proposition (20) 

        Let  𝑈 be a multiplication finitely generated faithful module  over good ring 𝑅  , 𝘈 is a WN-

prime ideal of 𝑅.Then 𝘈 𝑈 is a WN-prime submodule of  𝑈. 

Proof 

      Suppose that 0 ≠ 𝑎𝐻 ⊆ 𝘈 𝑈 where 𝑎 ∈ 𝑅,𝐻 is a submodule of  𝑈,implies that 0 ≠ 𝑎𝐼 𝑈 ⊆

𝐴 𝑈 for  𝑈 is a multiplication, it follows that 0 ≠ 𝑎𝐼 ⊆ 𝘈 + 𝑎𝑛𝑛( 𝑈).But  𝑈 is faithful, then 

𝑎𝑛𝑛( 𝑈) = (0). Thus 0 ≠ 𝑎𝐼 ⊆ 𝘈. But 𝘈 is a WN-prime ideal of 𝑅, then either 𝐼 ⊆ 𝘈 + 𝙹(𝑅) or 

𝑟 ∈ [𝘈 + 𝙹(𝑅):𝑅] = 𝘈 + 𝙹(𝑅). Hence 𝐼 𝑈 ⊆ 𝘈 𝑈 + 𝙹(𝑅) 𝑈 or 𝑟 𝑈 ⊆ 𝘈 𝑈 + 𝙹(𝑅) 𝑈. That is 

either 𝐼 𝑈 ⊆ 𝘈 𝑈 + 𝙹( 𝑈) or 𝑟 𝑈 ⊆ 𝘈 𝑈 + 𝙹( 𝑈). Thus either 𝐻 ⊆ 𝘈 𝑈 + 𝙹( 𝑈) or 𝑟 ∈

[𝘈 𝑈 + 𝙹( 𝑈):𝑅  𝑈]. Therefore 𝘈 𝑈 is a WN-prime submodule of  𝑈. 

 

Proposition (2.21) 

        Let  𝑈 be a finitely generated multiplication faithful module  over Artinian ring 𝑅,  and 𝐴 be  

a WN-prime ideal of 𝑅, then 𝐴 𝑈 is a WN-prime submodule of  𝑈. 

Proof 

      Similar as in Proposition (2.20). 

 

Proposition (2.22) 

         Let 𝘜 be a finitely generated projective multiplication  R-module , and 𝐴 is  a  WN-prime 

ideal  of 𝑅 with 𝑎𝑛𝑛(𝘜)⊆ 𝐴 then 𝐴𝘜 is  a WN-prime submodule of 𝘜. 

Proof 

      Suppose that 0 ≠ 𝑎𝑢 ∈ 𝐴𝘜 for 𝑎 ∈ 𝑅,𝑢 ∈ 𝘜 so, 0 ≠ 𝑎(𝑢) ⊆ 𝐴𝘜. Since 𝘜 is a multiplication, 

then (𝑢) = 𝙹𝘜 for some ideal 𝙹 of 𝑅, hence 0 ≠ 𝑎𝙹𝘜 ⊆ 𝐴𝘜, since 𝘜 is finitely generated 

multiplication, then  0 ≠ 𝑎𝙹 ⊆ 𝐴 + 𝑎𝑛𝑛(𝘜). But 𝑎𝑛𝑛(𝘜) ⊆ 𝐴, then 0 ≠ 𝑎𝙹 ⊆ 𝐴, since  𝐴 is  a 

WN-prime ideal of 𝑅 then by Corollary (2.4) either  𝙹 ⊆ 𝐴 + 𝙹(𝑅) or 𝑎 ∈ [𝐴 + 𝙹(𝑅):𝑅 𝑅] = 𝐴 +

𝙹(𝑅). That is either  𝙹𝘜 ⊆ 𝐴𝘜 + 𝙹(𝑅)𝘜 or 𝑎𝘜 ⊆ 𝐴𝘜 + 𝙹(𝑅)𝘜. But 𝘜 is a projective, then 

𝙹(𝑅) 𝑈 = 𝙹( 𝑈). Thus either  (𝑢) ⊆ 𝐴 𝑈 + 𝙹( 𝑈) or 𝑎 ∈ [𝐴 𝑈 + 𝙹( 𝑈):𝑅  𝑈]. That is either  

𝑢 ∈ 𝐴 𝑈 + 𝙹(𝘜) or 𝑎 ∈ [𝐴𝘜 + 𝙹(𝘜):𝑅 𝘜]. Thus 𝐴𝘜 is  a WN-prime submodule of 𝘜. 

 

Proposition (2.23) 

         Let  𝐻 be a WN-prime submodule of an R-module    𝑈,  then 𝑆−1𝐻 is a WN-prime 

submodule of 𝑆−1𝑅-module 𝑆−1 𝑈, where 𝑆 is a multiplicatively closed subset of 𝑅. 

 

Proof 

       Suppose that (0) ≠
𝑟1

𝑠1

𝑢

𝑠2
∈ 𝑆−1𝐻 for 

𝑟1

𝑠1
∈ 𝑆−1𝑅 and 

𝑢

𝑠2
∈ 𝑆−1 𝑈 and 𝑟1 ∈ 𝑅, 𝑠1,𝑠2 ∈ 𝑆,𝑢 ∈  𝑈. 

Then 
𝑟1𝑢

𝑡
∈ 𝑆−1𝐻, where 𝑡 = 𝑠1𝑠2 ∈ 𝑆, that is there exists non-zero element 𝑡1 ∈ 𝑆 such that 

0 ≠ 𝑡1𝑟1𝑢 ∈ 𝐻. But 𝐻 is a WN-prime submodule of   𝑈, then either 𝑡1𝑢 ∈ 𝐻 + 𝙹( 𝑈) or 𝑟1 ∈

[𝐻 + 𝙹( 𝑈):𝑅  𝑈], it follows that either 
𝑡1𝑢

𝑡1𝑠2
∈ 𝑆−1(𝐻 + 𝙹( 𝑈)) ⊆ 𝑆−1𝐻 + 𝙹(𝑆−1 𝑈)or 

𝑟1

𝑠1
∈



  

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𝑆−1[𝐻 + 𝙹( 𝑈):𝑅  𝑈] ⊆ [𝑆
−1𝐻 + 𝙹(𝑆−1 𝑈):𝑅 𝑆

−1 𝑈]. Hence either 
𝑢

𝑠2
∈ 𝑆−1𝐻 + 𝙹(𝑆−1 𝑈) or 

 
𝑟1

𝑠1
∈ [𝑆−1𝐻 + 𝙹(𝑆−1 𝑈):𝑅 𝑆

−1 𝑈]. Thus 𝑆−1𝐻 is a WN-prime submodule of 𝑆−1𝑅-module 𝑆−1 𝑈. 

        It is well known that if 𝜑 ∶  𝑈 ⟶ 𝑌 is an R-epimorphism and 𝐾𝑒𝑟𝜑 small submodule of R-

module   𝑈, then 𝜑( 𝙹( 𝑈)) = 𝙹(𝑌),𝜑−1( 𝙹(𝑌)) = 𝙹( 𝑈) [14, Cor. 9.1.5(a)]. 

 

Proposition (2.24) 

        Let  𝜑 ∶  𝑈 ⟶  𝑈′ be an 𝑅-epimorphism with 𝐾𝑒𝑟𝜑 is small submodule of 𝚄, and 𝐾 be a 

WN-prime submodule of  𝑈′,  then 𝜑−1(𝐾) is a WN-prime submodule of   𝑈. 

 

Proof  

       Let  0 ≠ 𝑟𝑥 ∈ 𝜑−1(𝐾) where 𝑟 ∈ 𝑅,𝑥 ∈  𝑈 with 𝑥 ∉ 𝜑−1(𝐾) + 𝙹( 𝑈), it follows that 

𝜑(𝑥) ∉  𝐾 + 𝜑( 𝙹( 𝑈)) = 𝐾 + 𝙹( 𝑈′). Since 0 ≠ 𝑟𝑥 ∈ 𝜑−1(𝐾), implies that 0 ≠ 𝑟 𝜑(𝑥) ∈ 𝐾. 

But 𝐾 be a WN-prime submodule of  𝑈′and 𝜑(𝑥) ∉ 𝐾 + 𝙹( 𝑈′), it follows that   𝑟 ∈

[𝐾 + 𝙹( 𝑈′):𝑅  𝑈
′], that is 𝑟 𝑈′ ⊆ 𝐾 + 𝙹( 𝑈′), hence 𝑟 𝜑( 𝑈) = 𝜑(𝑟 𝑈) ⊆ 𝐾 + 𝙹( 𝑈′). Implies 

that 𝑟 𝑈 ⊆ 𝜑−1(𝐾) + 𝙹( 𝑈). Therefore 𝜑−1(𝐾) is a WN-prime submodule of   𝑈. 

 

Proposition (2.25) 

        Let 𝑓 ∶ 𝘜 ⟶ 𝘜′ be an 𝑅-epimorphism with 𝐾𝑒𝑟𝑓 is small submodule of 𝘜, and 𝐻 be  a WN-

prime submodule of 𝘜 with 𝐾𝑒𝑟𝑓 ⊆ 𝐻. Then 𝑓(𝐻) is  a WN-prime submodule of 𝘜′. 

Proof 

       Since 𝐾𝑒𝑟𝑓 ⊆ 𝐻, that's clearly𝑓(𝐻) is a proper submodule of 𝘜′. Now, suppose that  0≠

𝑟𝑥′ ∈ 𝑓(𝐻), where 𝑟 ∈ 𝑅, 𝑥′ ∈ 𝘜′. Since 𝑓 is an epimorphism then 𝑓(𝑥) = 𝑥′ for some 𝑥 ∈ 𝘜, 

thus 0 ≠ 𝑟𝑥′ = 𝑟𝑓(𝑥) = 𝑓(𝑟𝑥) ∈ 𝑓(𝐻),it follows that there exists non-zero 𝑦 ∈ 𝐻  such that 

𝑓(𝑟𝑥) = 𝑓(𝑦), implies that  𝑓(𝑟𝑥 − 𝑦) = 0, hence  𝑟𝑥 − 𝑦 ∈ 𝐾𝑒𝑟 𝑓 ⊆ 𝐻 ⇒ 0 ≠ 𝑟𝑥 ∈ 𝐻. but 𝐻 is  

a WN-prime submodule of 𝘜, then either 𝑥 ∈ 𝐻 + 𝙹(𝘜) or 𝑟𝘜 ⊆ 𝐻 + 𝙹(𝘜), it follows that either 

𝑥′ = 𝑓(𝑥) ∈ 𝑓(𝐻) + 𝙹(𝘜′) or 𝑟 𝑈′ = 𝑟𝑓(𝘜) ⊆ 𝑓(𝐻) + 𝙹( 𝘜′). That is 𝑓(𝐻) is a WN-prime 

submodule of 𝘜′. 

 

3. Conclusion 

       In this article the concept WN-prime submodule was introduced and studied as generalization 

of a weakly prime submodule. The results that we set in this research are the following: 

1. Every weakly  prime submodule of R-module   𝑈 is WN-prime, but not conversely . 

2. A proper submodule  𝐻 of an R-module   𝑈 is a WN-prime if  and only if whenever  0 ≠

〈𝑟〉𝐿 ⊆ 𝐻 where 𝑟 ∈ 𝑅, 𝐿 is a submodule of  𝑈 implies that either 𝐿 ⊆ 𝐻 + 𝙹( 𝑈) or 

〈𝑟〉 𝑈 ⊆ 𝐻 + 𝙹( 𝑈) . 

3. A proper submodule  𝐻 of  an R-module   𝑈 is WN-prime if and only if [𝐻:𝑅 𝑥] ⊆

[𝐻 + 𝙹( 𝑈):𝑅  𝑈] ∪ [0:𝑅 𝑥] for all 𝑥 ∈  𝑈 and 𝑥 ∉ 𝐻 + 𝙹( 𝑈). 

4. Let 𝐻 be a proper submodule of an R-module   𝑈, with [𝐻 + 𝙹( 𝑈):𝑅  𝑈] is a  prime ideal of 

𝑅, then 𝐻 is  a WN-prime if and only if 𝐻(𝑆) ⊆ 𝐻 + 𝙹( 𝑈) for each  multiplicatively closed 

subset 𝑆  of 𝑅 with 𝑆 ∩ [𝐻 + 𝙹( 𝑈):𝑅  𝑈] = 𝜑. 

5. If a submodule 𝐻 of an R-module   𝑈 is small and 𝙹( 𝑈) is a weakly prime submodule of  𝑈, 

then 𝐻 is  WN-prime submodule of  𝑈. 



  

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6. Let  𝑈 be a multiplication module  over Artinian ring 𝑅, and  𝐻 is a WN-prime submodule of 

 𝑈 then [𝐻:𝑅  𝑈] is  a WN-prime ideal of 𝑅. 

7. If  𝑈 is a projective multiplication  R-module , and  𝐻 is a WN-prime submodule of  𝑈 then 

[𝐻:𝑅  𝑈] is a  WN-prime ideal of 𝑅. 

8. If   𝑈 is finitely generated faithful multiplication module  over good ring 𝑅,  and 𝐴 be  WN-

prime ideal of 𝑅, then 𝐴 𝑈 is WN-prime submodule of  𝑈. 

9. If   𝑈 is finitely generated projective multiplication  R-module  then 𝐴 𝑈 is a WN-prime 

submodule of  𝑈 for all WN-prime ideal 𝐴 of 𝑅 with 𝑎𝑛𝑛( 𝑈) ⊆ 𝐴. 

10. If 𝐻 is a WN-prime submodule  of an R-module   𝑈, then 𝑆−1𝐻 is  a WN-prime submodule 

of 𝑆−1𝑅-module 𝑆−1 𝑈, where 𝑆 is a multiplicatively closed subset of 𝑅. 

 

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