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Mathematics | 109 

Ibn Al-Haitham Jour. for Pure & Appl. Sci.                                                   IHJPAS      
                            https://doi.org/10.30526/31.3.2000                                                                       Vol. 31 (3) 2018 

WE-Prime Submodules and WE-Semi-Prime Submodules 

Saif A. Hussin 
Haibt K. Mohammadali 

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

Saif.a19881988@gmail.com  

    
Article history: Received 5 August 2018, Accepted 18 September 2018, Published December 

2018"" 

Abstract 
"     In this article, "we introduce the concept of a WE-Prime submodule", as a stronger form 
of a weakly prime submodule. And as a "generalization of WE-Prime submodule, we 
introduce the concept of WE-Semi-Prime submodule, which is also a stronger form of a 
weakly semi-prime submodule. Various basic properties of these two concepts are discussed. 
Furthermore, the relationships between "WE-Prime submodules and weakly prime 
submodules"and studied. On the other hand, the relation between WE-Prime submodules and 
WE- Semi - Prime submodules are consider."Also"the relation of "WE – Sime - Prime 
submodules and weakly semi-prime submodules" are explained. Behind that, some 
characterizations of these concepts are investigated". 
 
Keywords: "weakly prime submodules, weakly semi-prime submodules, "WE-Prime 
submodules, WE-Semi-Prime submodules. 

1. Introduction  
""Weakly prime submodule" "have been introduced and studied" by Hadi M. A in [1], where 
"a proper submodule K of an R-module X is called a weakly prime, if" wherever 0 𝑟𝑥 ∈ 𝐾, 
where "𝑟 ∈ 𝑅, 𝑥 ∈ 𝑋", implies that either 𝑥 ∈ 𝐾 or 𝑟 ∈ 𝐾: 𝑋 , where 𝐾: 𝑋 𝑎 ∈ 𝑅 ∶ 𝑎𝑋
𝐾 . "Weakly semi-prime submodule have been introduced and studied by Farzalipour F in 
[2], "where a proper submodule K of an R-module" X "is called a weakly semi-prime if" 
wherever 0 𝑟 𝑥 ∈ 𝐾, where 𝑟 ∈ 𝑅, 𝑥 ∈ 𝑋, implies that 𝑟𝑥 ∈ 𝐾. ""Throughout this note all 
rings will be commutative with identity, and all R-modules are left unitary"". "A proper 
submodule K of an R-module X is said to be fully invariant if 𝑓 𝐾 𝐾  for each 𝑓 ∈
𝐸𝑛𝑑 𝑋  [3]. An R-module M is called X- Injective", if for "every R-homomorphism       
𝑔: 𝑁 ⟶ 𝑀 ", and every R-homomorphism 𝑓: 𝑁 ⟶ 𝑋 , there exists an R-homomorphism 
ℎ: 𝑋 ⟶ 𝑀, "where N is an R-module" such that ℎ𝑜𝑓 𝑔 [5]". "An R-module P is called X-
Projective if for every R-homomorphism 𝑓: 𝑃 ⟶ 𝑁  and every R-epimorphism 𝑔: 𝑀 ⟶ 𝑁 , 
there exists an R-homomorphism ℎ: 𝑃 ⟶ 𝑀 such that 𝑔𝑜ℎ 𝑓 [5]. An R-module X is called 
a scalar module" "if for each" 𝑓 ∈ 𝐸𝑛𝑑 𝑋 , "there exists 𝑟 ∈ 𝑅 such that 𝑓 𝑚 𝑟𝑚 for each 
𝑚 ∈ 𝑋 [6]". 
 

2. WE-Prime Submodules" 
   In this section, we introduce the concept WE-Prime submodule as a stronger form of a 
weakly prime submodule, and established some of its basic properties, examples and 
characterizations. 



 

Mathematics | 110 

Ibn Al-Haitham Jour. for Pure & Appl. Sci.                                                   IHJPAS      
                            https://doi.org/10.30526/31.3.2000                                                                       Vol. 31 (3) 2018 

Definition (1) 
   A proper submodule K of an R-module X is said to be a weakly endo-prime (for a short 
WE-Prime), where 𝐸 𝐸𝑛𝑑 𝑋 , if wherever, 0 𝜓 𝑥 ∈ 𝐾 , where 𝜓 ∈ 𝐸𝑛𝑑 𝑋 , 𝑥 ∈ 𝑋 , 
implies that either 𝑥 ∈ 𝐾 or 𝜓 𝑥 𝐾. "And an ideal I of a ring R is said to be a weakly 
endo-prime ideal" (WE-Prime ideal), "if I is a WE-Prime as an R-submodule of an R-module 
R". 
""The following" proposition gives relation of "WE-Prime submodules and weakly prime 
submodules". 

Proposition (2)  
"Every WE-Prime submodule of an R-module" X" is a weakly prime submodule of X". 
Proof" 
" Assume that K is a WE-Prime submodule of X, and" 0 𝑟𝑥 ∈ 𝐾, where 𝑟 ∈ 𝑅, 𝑥 ∈ 𝑋, with 
𝑥 ∉ 𝐾". "Now, let 𝜓: 𝑋 ⟶ 𝑋 be a mapping defined by 𝜓 𝑥 𝑟𝑥 for all 𝑥 ∈ 𝑋. Clearly 𝜓 ∈
𝐸𝑛𝑑 𝑋 . In fact we have 0 𝑟𝑥 𝜓 𝑥 ∈ 𝐾". ""But K is a WE-Prime submodule of X, and 
𝑥 ∉ 𝐾", implies that 𝜓 𝑥 𝐾, hence 𝑟𝑥 𝐾, so 𝑟 ∈ 𝐾: 𝑋 . "Therefore K is a weakly prime 
submodule of X". 
The converse of Proposition (2)" "is not true in general, as the following example shows". 

Example (3)" 
" Let 𝑋 𝑍 ⨁ 𝑍" and R=Z, 𝐾 〈0〉⨁3𝑍. Clearly K "is a weakly prime submodule of X, but 
K is not WE-Prime submodule of X". Since we define 𝜓: 𝑋 ⟶ 𝑋 "by 𝜓 𝑎, 𝑏 0, 𝑏  for all 
𝑎, 𝑏 ∈ 𝑋 ". Clearly 𝜓 ∈ 𝐸𝑛𝑑 𝑋 . Now 0, 0 𝜓 1, 3 0, 3 ∈ 𝐾 , but 1, 3 ∉ 𝐾  and 

𝜓 𝑋 0 ⨁𝑍 ≰ 𝐾". 
""The converse of Proposition (2)" "is true in the class of cyclic R-modules, as the following 
proposition shows". 

Proposition (4)" 
"   Let X be a cyclic R-module, and" K is a "proper submodule of X such that K is a weakly 
prime submodule of X."Then K is a WE-Prime submodule of X". 
Proof" 
   "Assume that K is a weakly prime submodule of cyclic R-module X", "where 𝑋 𝑅𝑚, 𝑚 ∈
𝑋". "Suppose that 0 𝜓 𝑥 ∈ 𝐾, where 𝜓 ∈ 𝐸𝑛𝑑 𝑋 , 𝑥 ∈ 𝑋 and 𝑥 ∉ 𝐾". "Now, let 𝑦 ∈ 𝑋, 
then 𝑦 𝑟𝑚 and 𝑥 𝑟 𝑚 for some 𝑟, 𝑟 ∈ 𝑅". "Thus, 0 𝜓 𝑥 𝑟 𝜓 𝑚 ∈ 𝐾, but K is a 
weakly prime submodule of X, then either 𝑟 ∈ 𝐾: 𝑋  or 𝜑 𝑚 ∈ 𝐾". "But 𝑟 ∉ 𝐾: 𝑋  for 
𝑥 𝑟 𝑚 ∉ 𝐾. Hence 𝜓 𝑚 ∈ 𝐾, hence 𝜓 𝑦 𝑟𝜓 𝑚 ∈ 𝐾. Therefore 𝜓 𝑋 𝐾". 
 
Corollary (5) 
    Let K be a proper submodule of a cyclic R-module X". "Then K is a WE-Prime if and only 
if K is a weakly prime submodule of X. 
 
Proposition (6) 
    Let X be a faithful R-module", "and K is a WE-Prime submodule of X". "Then 𝐾: 𝑋  is a 
WE-Prime ideal of R. 
 
 



 

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Ibn Al-Haitham Jour. for Pure & Appl. Sci.                                                   IHJPAS      
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Proof" 
   Since K is a WE-Prime submodule of X", "then by Proposition (2.2), K is a weakly prime 
submodule of X". "Hence by [1, Prop.2.4]", we get 𝐾: 𝑋  "is a weakly prime ideal of R. But 
R is a cyclic R-module", "then by Proposition (2.4), we get 𝐾: 𝑋  is a       WE-Prime ideal of 
R". 
"We need to recall the following result before we introduce the next proposition". 
 
Lemma (7) [3] 
"Let N and K be" "two submodules of an R-module X, then 

1. "If 𝑁 𝐾, then 𝑁: 𝑋 𝐾: 𝑋 ". 
2. "If 𝑁 𝐾, then 𝑁: 𝑋 𝑁: 𝐾 ". 

""The following proposition is a characterization of a WE-Prime submodules". 

Proposition (8)" 
     Let K be a proper fully invariant submodule of an R-module X". "Then K is a WE-Prime 
submodule of X if and only if" 𝐾: 𝜓 𝑋 𝐾: 𝜓 𝐻  for all 𝜓 ∈ 𝐸𝑛𝑑 𝑋  and "a non-zero 
submodule H of X with 𝐾 𝐻". 

Proof" 
      ⟹  Assume that K is a WE-Prime submodule of X, and" H "is a non-zero submodule of 
X such that 𝐾 𝐻 ". "Let 𝜓 ∈ 𝐸𝑛𝑑 𝑋 , then by Lemma (2.7)(2) we have 𝐾: 𝜓 𝑋
𝐾: 𝜓 𝐻 , since 𝐾 𝐻, "then there exists 𝑥 ∈ 𝐻 and 𝑥 ∉ 𝐾". "Now, "suppose that b is a 

non-zero element in" 𝐾: 𝜓 𝐻 , then 0 𝑏𝜓 𝐻 𝐾, implies that 0 𝑏𝜓 𝑥 ∈ 𝐾, where 
𝑥 ∈ 𝐻 𝑋". "Define 𝜓: 𝑋 ⟶ 𝑋 by 𝜓 𝑦 𝑏𝜓 𝑦  for all 𝑦 ∈ 𝑋, clearly 𝜓 ∈ 𝐸𝑛𝑑 𝑋 , also 
0 𝑏𝜓 𝑥 𝜓 𝑥 ∈ 𝐾. But K is a WE-Prime submodule of X, and 𝑥 ∉ 𝐾, then 𝜓 𝑋 𝐾, 
implies that 𝑏𝜓 𝑋 𝐾  and hence 𝑏 ∈ 𝐾: 𝜓 𝑋 . Thus 𝐾: 𝜓 𝐻 𝐾: 𝜓 𝑋 , and it 
follows that  𝐾: 𝜓 𝑋 𝐾: 𝜓 𝐻 ". 
" ⟸  Assume that  0 𝜓 𝑥 ∈ 𝐾", "where 𝑥 ∈ 𝑋 and 𝜓 ∈ 𝐸𝑛𝑑 𝑋 , and suppose that 𝑥 ∉
𝐾", we want to show that 𝜓 𝑋 𝐾. Since 𝑥 ∉ 𝐾, then 𝐾 𝐾 𝑅𝑥, where 𝐾 𝑅𝑥 "is a 
non-zero submodule of X". Thus by our hypothesis, we get 𝐾: 𝜓 𝑋 𝐾: 𝜓 𝐾 𝑅𝑥 . 
Since K is a fully invariant, then 𝜓 𝐾 𝐾 and "𝜓 𝑅𝑥 𝐾, it follows that 𝜓 𝐾 𝑅𝑥
𝐾". Hence 𝐾: 𝜓 𝐾 𝑅𝑥 𝑅, therefore 1 ∈ 𝐾: 𝜓 𝐾 𝑅𝑥 , implies that 1 ∈ 𝐾: 𝜓 𝑋 , 
hence 𝜓 𝑋 𝐾. Thus K is a WE-Prime submodule of X". 

Proposition (9) 
""Let X be an R-module, and L", H are "submodules of X, with H is a fully invariant 

submodule of X and 𝐻 𝐿". "If  is a WE-Prime submodule of  , then L is a WE-Prime 

submodule of X". 
 
Proof 
"Assume that 0 𝜓 𝑥 ∈ 𝐿, where 𝑥 ∈ 𝑋 and 𝜓 ∈ 𝐸𝑛𝑑 𝑋 . If 𝑥 ∉ 𝐿, then we must show that 

𝜓 𝑋 𝐿. Define 𝜓 : ⟶  by 𝜓 𝑥 𝐻 𝜓 𝑥 𝐻 for all 𝑥 ∈ 𝑋. To prove that 𝜑  is 

well define, suppose that 𝑥 𝐻 𝑥 𝐻  where 𝑥 , 𝑥 ∈ 𝑋 , then 𝑥 𝑥 ∈ 𝐻 , hence  
𝜓 𝑥 𝑥 ∈ 𝜓 𝐻 𝐻 because H is a fully invariant. It follows that 𝜓 𝑥 𝜓 𝑥 ∈ 𝐻. 
Hence 𝜓 𝑥 𝐻 𝜓 𝑥 𝐻, implies that 𝜓 𝑥 𝐻 𝜓 𝑥 𝐻. Since 0 𝜓 𝑥 ∈



 

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𝐿, implies that  0 𝜓 𝑥 𝐻 𝜓 𝑥 𝐻 ∈
𝐿
𝐻

. But   is a WE-Prime submodule of , and 

𝑥 𝐻 ∉
𝐿
𝐻

, implies that 𝜓
𝑋
𝐻

𝐿
𝐻

 , thus, we have 
𝐿
𝐻

, it follows that 𝜓 𝑋 𝐻

𝐿. Thus   𝜓 𝑋 𝐿". Hence "L is a WE-Prime submodule of X". 
Proposition (10)" 
  "Let L and K are submodules of an R-module X", "with L is an X-injective, and K is a WE-
Prime submodule of X". "Then either 𝐿 𝐾or 𝐾 ∩ 𝐿 is a WE-Prime submodule of L". 
 
Proof 
   "Assume that 𝐿 ≰ 𝐾", "then 𝐾 ∩ 𝐿 is a proper submodule of L". Now, let   0 𝜓 𝑥 ∈ 𝐾 ∩
𝐿, where 𝑥 ∈ 𝐿  and 𝜓 ∈ 𝐸𝑛𝑑 𝐿 . Suppose that 𝑥 ∉ 𝐾 ∩ 𝐿 , then 𝑥 ∉ 𝐾 . Now, consider the 
following diagram, "where 𝔦 is the inclusion map. Since L is an X-injective then there exists" 
𝜙: 𝑋 ⟶ 𝐿 such that 𝜙𝑜𝔦 𝜓. Clearly 𝜙 ∈ 𝐸𝑛𝑑 𝑋 , but 0 𝜓 𝑥 𝜙𝑜𝔦 𝑥 𝜙 𝑥 ∈ 𝐾, 
implies that 0  𝜙 𝑥 ∈ 𝐾. But K is a   WE-Prime submodule of X and 𝑥 ∉ 𝐾, then 𝜙 𝑋
𝐾. "Also, we have    𝜓 𝐿 𝜙𝑜𝔦 𝐿 𝜙 𝐿 𝐿 and 𝜓𝜓 𝐿 𝜙 𝐿 𝜙 𝑋 𝐾. Hence   
𝜓 𝐿  𝐾 ∩ 𝐿, it follows that 𝐾 ∩ 𝐿 is a WE-Prime submodule of L". 
 
Proposition (11)" 
"Let X be an R-module"and K, L are non-trivial submodules of X such that L is a WE-Prime 
submodule of X"and IK is a non-zero submodule of L for some ideal I of R. If 𝐼 𝐿: 𝑋  then  
𝐾 𝐿". 
 
Proof 
"Suppose that 𝑦 ∈ 𝐾 , since 𝐼 ≰ 𝐿: 𝑋 , then there exists 𝑖 ∈ 𝐼  and 𝑖 ∉ 𝐿: 𝑋 ". "Now, let 
𝜓: 𝑋 ⟶ 𝑋 define by 𝜓 𝑥 𝑖𝑥 for all submodule 𝑥 ∈ 𝑋, clearly 𝜓 ∈ 𝐸𝑛𝑑 𝑋 ". "Since IK "is 
a non-zero submodule of L", "then iy is a non-zero element in K". That is            0 𝜓 𝑦
𝑖𝑦 ∈ 𝐼𝐾 𝐿 , implies that 0 𝑖𝑦 ∈ 𝐿 , but "L is a WE-Prime submodule of X, and 𝑖𝑋
𝜓 𝑋 ≰ 𝐿, implies that 𝑦 ∈ 𝐿. Thus 𝐾 𝐿". 
 
Proposition (12) 
"Let X be an R-module and 𝜓: 𝑋 ⟶ 𝑋 be an R-homomorphism", "and K be a proper fully 
invariant WE-Prime submodule of X with 𝜓 𝑋 ≰ 𝐾. Then 𝜓 𝐾  is a        WE-Prime 
submodule of X". 
 
Proof 
"Clearly 𝜓 𝐾  is a proper submodule of" X. Now, assume that 0 𝜙 𝑥 ∈ 𝜓 𝐾  where 
𝑥 ∈ 𝑋, 𝜙 ∈ 𝐸𝑛𝑑 𝑋 . If  𝑥 ∉ 𝜓 𝐾 , then 𝜓 𝑥 ∉ 𝐾, "it follows that" 𝑥 ∉ 𝐾 "because K is a 
fully invariant submodule of X". "We must prove that 𝜙 𝑋 𝜓 𝐾 . Since 0 𝜓𝑜𝜙 𝑥
𝜓 𝜙 𝑥 ∈ 𝐾. "That is 0 𝜓 𝜙 𝑥 ∈ 𝐾". "But K is a WE-Prime submodule of X, and 𝑥 ∉
𝐾", "it follows that 𝜓𝑜𝜙 𝑋 𝐾, implies that           𝜙 𝑋 𝜓 𝐾 . Hence 𝜓 𝐾  is a 
WE-Prime submodule of X". 
3. WE-Semi-Prime Submodules" 
"In this section, we introduce the" concept "of WE-Semi-Prime submodule as a generalization 
of" a WE-Prime "submodule and" stronger form of a weakly semi-prime "submodule and give 
some basic properties", "examples and characterizations of this concept". 



 

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Definition (13)" 
"A proper submodule K of an R-module" X "is said to be a weakly endo semi-prime 
submodule of X (for a short WE-Semi-Prime)", where 𝐸 𝐸𝑛𝑑 𝑋 , if, wherever 0
ψ 𝑥 ∈ 𝐾, where 𝑥 ∈ 𝑋 and ψ ∈ 𝐸𝑛𝑑 𝑋 ", "implies that ψ 𝑚 ∈ 𝐾. "And an ideal I of a 
ring R is said to be a" weakly endo semi-"prime ideal of R, if I is a" weakly endo semi-"prime 
as an R-submodule of R-module R". 
 
Proposition (14)  
"Every WE-Prime submodule of an R-module X" "is a WE-Semi-Prime submodule of X". 
 
Proof 
"Let K be a WE-Prime submodule of X", "and 0 ψ 𝑥 ∈ 𝐾", where 𝑥 ∈ 𝑋,  ψ ∈ 𝐸𝑛𝑑 𝑋 . 
Since "K is a WE-Prime submodule, and  0 ψ ψ 𝑥 ∈ 𝐾 ", then "either ψ 𝑥 ∈ 𝐾  or 
ψ 𝑋 𝐾". "Thus in any case ψ 𝑥 ∈ 𝐾". "Hence K is a WE-Semi-Prime submodule of X". 
""The converse of Proposition (3.2) is not true in general", "as the following example shows 
that". 
Example (15)"  
"Let X=Z and R=Z, K=10Z as a" Z-module of X. "Then K is a WE-Semi-Prime but not WE-
Prime submodule of X, since if we defined ψ: 𝑍 ⟶ 𝑍 by ψ 𝑥 𝑥, ψ ∈ 𝐸𝑛𝑑 𝑋  and 0
2ψ 5 10 ∈ 𝐾 , but 5 ∉ 𝐾  and ψ 𝑍 𝑍 ≰ 𝐾 10𝑍 , hence K is not   WE-Prime 
submodule of X. But K is a WE-Semi-Prime, since 0 ψ 10 ψ ψ 10 10 ∈ 𝐾 , 
implies that  ψ 10 10 ∈ 𝐾". 
 
Proposition (16)  
"Every WE-Semi-Prime submodule of an R-module X" "is a weakly semi-prime submodule 
of X". 
Proof 
"Let K be a WE-Semi-Prime submodule of" X, "and" 0 𝑟 𝑥 ∈ 𝐾 , where 𝑟 ∈ 𝑅, 𝑥 ∈ 𝐾 . 
Now, let ψ: 𝑋 ⟶ 𝑋 defined by ψ 𝑥 𝑟𝑥  for all 𝑥 ∈ 𝑋, clearly ψ ∈ 𝐸𝑛𝑑 𝑋 ". "Now, 0
𝑟 𝑥 ψ 𝑥 ∈ 𝐾, but "K is a WE-Semi-Prime submodule of" X, "implies that ψ 𝑥 𝑟𝑥 ∈
𝐾". "Thus K is a weakly semi-prime submodule of X". 
""The converse of Proposition" "(3.4) is not true in general, as the following example shows". 
 
Example (17)" 
"Let 𝑋 𝑍⨁𝑍, R"=Z, 𝐾 𝑍⨁10𝑍, K "is a weakly semi-prime submodule of X" but not 
WE-Semi-Prime : Let 𝑟 2 ∈ 𝑍  and 𝑥 3,5 ∈ 𝑋 , then  0 2 3,5 12,20 ∈ 𝐾 , 
implies that 2 3,5 6,10 ∈ 𝐾. To show that K is not WE-Semi-Prime : Let ψ: 𝑋 ⟶ 𝑋 
"defined by ψ 𝑥, 𝑦 𝑦, 𝑥  for all 𝑥, 𝑦 ∈ 𝑍 ". Clearly ψ ∈ 𝐸𝑛𝑑 𝑋 . Now, take ψ 0,5
5,0 ∉ 𝐾  but ψ 0,5 ψ ψ 0,5 ψ 5,0 0,5 ∈ 𝐾 . Hence K is not WE-Semi-

Prime submodule of X". 
 
Proposition (18)" 
"Let K be a submodule of an R-module" X with 𝐾 ∩∝∈∧ 𝐿∝, where each 𝐿∝ "is a  WE-Prime 
submodule of X. "Then K is a WE-Semi-Prime submodule of X. 

Proof" 
"Suppose that" 0 ψ 𝑥 ∈ 𝐾, where 𝑥 ∈ 𝑋, ψ ∈ 𝐸𝑛𝑑 𝑋 , then 0 ψ 𝑥 ∈ 𝐿∝ for each ∝
∈∧. But 𝐿∝ "is a WE-Prime submodule of X, hence by Proposition (3.2)" 𝐿∝ is a WE-Semi-



 

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Prime. "Thus ψ 𝑥 ∈ 𝐿∝ for each ∝∈∧. Therefore ψ 𝑥 ∈  ∩∝∈∧ 𝐿∝. Hence K is a WE-Semi-
Prime submodule of X. 
"The following proposition shows that in the class of scalar modules, weakly semi-prime 
submodule and WE-Semi-Prime submodules are coinciding. 
 
Proposition (19)" 
"Let X be a scalar module, and L is a proper submodule of X. Then L is a WE-Semi-Prime 
submodule of X, if and only if L is a weakly semi-prime submodule of X. 
Proof 
" ⟹  "Follows from Proposition (3.4). 
" ⟸  Suppose that L is a weakly semi-prime submodule of X", and 0 𝜙 𝑥 ∈ 𝐿, where 
𝑥 ∈ 𝑋  and 𝜙 ∈ 𝐸𝑛𝑑 𝑋 . Since X is a scalar module, "then there exists 𝑟 ∈ 𝑅  such that 
𝜙 𝑥 𝑟𝑥 for each 𝑥 ∈ 𝑋". "Now", 0 𝜙 𝑥 𝜙 𝜙 𝑥 𝜙 𝑟𝑥 𝑟 𝑥 ∈ 𝐿. But L "is a 
weakly semi-prime submodule of X", implies that 𝑟𝑥 ∈ 𝐿. "Hence 𝜙 𝑥 ∈ 𝐿". "Thus L is a 
WE-Semi-Prime submodule of X". 
""The following propositions are characterizations of WE-Semi-Prime submodules". 

 
Proposition (20)"   
"Let X be an R-module, and" L is "a proper submodule of X". Then L is a WE-Semi-Prime 
submodule" if and only "if 0 𝜙 𝐾 𝐿, where K is a submodule of X and 𝜙 ∈ 𝐸𝑛𝑑 𝑋 , 
implies that 𝜙 𝐾 𝐿". 
Proof 
" ⟹  Assume that 0 𝜙 𝐾 𝐿", where "K is a submodule of" X, 𝜙 ∈ 𝐸𝑛𝑑 𝑋 , implies 
"that 0 𝜙 𝑥 ∈ 𝐿 for all 𝑥 ∈ 𝐾 𝑋". "Since L is a WE-Semi-Prime submodule of X, then 
𝜙 𝑥 ∈ 𝐿 for all 𝑥 ∈ 𝑋. Thus 𝜙 𝐾 𝐿. 
" ⟸  Suppose that 0 𝜙 𝑥 ∈ 𝐿, where 𝑥 ∈ 𝑋, and 𝜙 ∈ 𝐸𝑛𝑑 𝑋 , then by hypothesis, we 
have 𝐾 𝑥  is a submodule of X, and 0 𝜙 𝐾 ∈ 𝐿, implies that 𝜙 𝐾 𝐿, "it follows 
that 𝜙 𝑥 ∈ 𝐿. "Hence L is a WE-Semi-Prime submodule of X. 
 
Proposition (21)"  
"Let X be an R-module, and" L is "a proper submodule of X". "Then L is a WE-Semi-Prime 
submodule of X, if and only if", "wherever 0 𝜙 𝑥 ∈ 𝐿, 𝑥 ∈ 𝑋, 𝜙 ∈ 𝐸𝑛𝑑 𝑋 , and for 𝑛
2, implies that 𝜙 𝑥 ∈ 𝐿". 
Proof 
" ⟹  Follows by inducation on 𝑛 ∈ 𝑍 ". 
" ⟸  Direct from definition of WE-Semi-Prime submodule". 
"In the class of scalar module, we get the following characterizations of WE-Semi-Prime 
submodules. 
 
Proposition (22) 
""Let X be a scalar R-module, and L be a proper submodule of X. Then the following 
statements are equivalent:" 

1. "L is a WE-Semi-Prime submodule of X". 
2. " 𝐿: 𝑟 0 : 𝑟 ∪ 𝐿: 𝑟  for non-zero r in R". 
3. 𝐿: 𝑟 0 : 𝑟  or 0 : 𝑟 𝐿: 𝑟  for non-zero r in R". 

 



 

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Proof 
" 1 ⟹ 2  Since L is a WE-Semi-Prime submodule of X, "then by Proposition (3.4) L is a 
weakly semi-prime submodule of X. Now, let 𝑥 ∈ 𝐿: 𝑟 , implies that 𝑟 𝑥 ∈ 𝐿, either 0
𝑟 𝑥 ∈ 𝐿  or 𝑟 𝑥 0 . If  0 𝑟 𝑥 ∈ 𝐿 , implies that 𝑟𝑥 ∈ 𝐿 , hence 𝑥 ∈ 𝐿: 𝑟 . If  𝑟 𝑥 0 , 
implies that 𝑥 ∈ 0 : 𝑟 , hence, we get 𝐿: 𝑟 𝐿: 𝑟 ∪ 0 : 𝑟 . Clearly we have by 
Lemma (2.7), 𝐿: 𝑟 𝐿: 𝑟 , and 0 : 𝑟 𝐿: 𝑟 , hence 𝐿: 𝑟 ∪ 0 : 𝑟 𝐿: 𝑟 . 
Thus the equality holds". 
" 2 ⟹ 3  Direct". 
" 3 ⟹ 1  To prove first L is a weakly semi-prime submodule of X". "Suppose that"  0
𝑟 𝑥 ∈ 𝐿, where 𝑥 ∈ 𝑋,  𝑟 ∈ 𝑅, implies that 𝑥 ∈ 𝐿: 𝑟  and 𝑥 ∉ 0 : 𝑟 . Thus by hypothesis, 
we get 𝑥 ∈ 𝐿: 𝑟 , implies that 𝑟𝑥 ∈ 𝐿, hence L "is a weakly semi-prime submodule of X". 
"Thus by Proposition (3.7), we have L is a WE-Semi-Prime submodule of X". 
Recall that "an element x in R-module X is called" torsion if                        
0 𝑎𝑛𝑛 𝑥 𝑟 ∈ 𝑅 ∶ 𝑟𝑥 0 . The set of all torsion elements denoted by T(X), which is a 
submodule of X. If T(X)=(0), then X is called torsion free [3]. 
 
Proposition (23)  
"Let X is a torsion free scalar R-module, and L be a proper submodule of X, such that L is a 
WE-Semi-Prime submodule of X. Then 𝐿: 𝐼  is a WE-Semi-"Prime submodule of X for any 
non-zero ideal I of R". 
Proof 
"Since L is a WE-Semi-Prime submodule of X", "then by Proposition (3.4) L is a weakly 
semi-prime submodule of X". "Thus by [2, Prop.27] we get 𝐿: 𝐼  is a weakly semi-prime 
submodule of X. "But X is a scalar module, hence by Proposition (3.7), we have 𝐿: 𝐼  is a 
WE-Semi-Prime submodule of X". 
 
Proposition (24) 
   Let 𝜙: 𝑋 ⟶ 𝑋′ be an R-epimorphism, and" L is a WE-Semi-Prime submodule of X with 
𝐾𝑒𝑟𝜙 𝐿. "Then 𝜙 𝐿  is a WE-Semi-Prime submodule of X', where X' is an X-projective R-

module. 
Proof 
  Clearly 𝜙 𝐿  is a proper submodule of X'. Assume that 0 𝑓 𝑥′ ∈ 𝜙 𝐿  where 𝑥′ ∈ 𝑋′,    
and 𝑓 ∈ 𝐸𝑛𝑑 𝑋′ , we prove that 𝑓 𝑥′ ∈ 𝜙 𝐿 , since 𝜙 is an epimorphism, and 𝑥′ ∈ 𝑋′, then 
there exists 𝑥 ∈ 𝑋  such that 𝜙 𝑥 𝑥′ . "Consider the following diagram since X' is X-
projective", then there exists a homomorphism h such that 𝜙oh f . Now, 0 𝑓 𝑥′
𝑓 𝑓 𝑥′ ∈ 𝜙 𝐿 , "implies that 0 𝜙 ∘ h ∘ 𝜙 ∘ h 𝑥′ ∈ 𝜙 𝐿 , and hence 0 𝜙 h ∘ 𝜙 𝑥 ∈
𝜙 𝐿 . But 𝐾𝑒𝑟𝜙 𝐿," "then 0 h ∘ 𝜙 𝑥 ∈ 𝐿. Since L is a WE-Semi-Prime submodule 
of X, then 𝜙 ∘ h 𝑥 , implies that 𝜙 h ∘ 𝜙 𝑥 ∈ 𝜙 𝐿  hence 𝜙 ∘ h 𝜙 𝑥 ∈ 𝜙 𝐿  implies 
that 𝑓 𝑥′ ∈ 𝜙 𝐿 . Therefore 𝜙 𝐿  is a WE-Semi-Prime submodule of X'. 
"As a direct consequence of Proposition (3.12) we get the following corollary. 
 
Corollary (25)" 
"Let L and K be a submodule of an R-module X with 𝐾 𝐿, "and L is a WE-Semi-Prime 

submodule of X. "Then  is a WE-Semi-Prime submodule of , where  is an X-projective R-

module. 



 

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"Recall that an R-module X is multiplication if every submodule K of X is of the form K=IX 
for some ideal I of R [7]. 
 
Proposition (26)" 
"Let X be a multiplication R-module and L is a weakly semi-prime submodule of X"," then L 
is a WE-Semi-Prime submodule of X". 

Proof" 
"Suppose that 0 𝑓 𝑥 ∈ 𝐿, where 𝑥 ∈ 𝑋, 𝑓 ∈ 𝐸𝑛𝑑 𝑋 ."Since X is a multiplication, then by 
[8, Coro.1.2] there exists 𝑠 ∈ 𝑅 such that 𝑓 𝑥 𝑠𝑥 for all 𝑥 ∈ 𝑋". "Hence 0 𝑓 𝑓 𝑥
𝑠 𝑥 ∈ 𝐿. But L is a weakly semi-prime, implies that 𝑠𝑥 ∈ 𝐿. Thus  𝑓 𝑥 ∈ 𝐿, so L is a WE-
Semi-Prime submodule of X. 
"It is well-known every cyclic R-module is a multiplication [7], we get the following result. 

Corollary (27)" 
"Let X be a cyclic R-module, and L is a proper submodule of X. "Then L is a WE-Semi-
Prime submodule if and only if L is a weakly semi-prime. 
We end this section by the following result. 
 
Proposition (28)"  
"Let X be a faithful multiplication R-module, and L is a proper submodule of X. "Then L is a 
WE-Semi-Prime submodule of X if and only if 𝐿: 𝑋  is a WE-Semi-Prime ideal of R". 
Proof 
" ⟹  Since L is a WE-Semi-Prime submodule of X, "then by Proposition (3.4) L is a weakly 
semi-prime submodule of X". "Hence by [2, Prop.29], we have 𝐿: 𝑋  is a weakly semi-prime 
ideal of R. "Therefore 𝐿: 𝑋  is a weakly semi-prime as R-submodule of R-module R". "But R 
is cyclic R-module, implies that by Corollary (27)  𝐿: 𝑋  is a WE-Semi-Prime R-submodule 
of R-module R. Hence 𝐿: 𝑋  is a WE-Semi-Prime ideal of R. 
" ⟸  Since  𝐿: 𝑋  is a WE-Semi-Prime ideal of R, "implies that  𝐿: 𝑋  is a weakly semi-
prime ideal of R". "Hence by [2, Theo.30] we have L is a weakly semi-prime submodule of X. 
But X is a multiplication, then by Proposition (26) L is a WE-Semi-Prime submodule of X. 
""As a direct consequence of Proposition (27), we get the following result". 
 
Corollary (3.17)" 
"Let X be a faithful cyclic R-module", "and L is a proper submodule of X. "Then L is a WE-
Semi-Prime submodule of X if and only if 𝐿: 𝑋  is a WE-Semi-Prime ideal of R.  
 
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