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Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (3) 2019 

 

 

  

 

 

 

           Ali Sh. Ajeel                                               Haibat  K. Mohammadali        

 

 

 

 

 

 

 

Abstract  

    We introduce in this paper the concept of approximaitly semi-prime submodules of unitary 

left 𝑅-module 𝑇 over a commutative ring 𝑅 with identity as a generalization of a prime 

submodules and semi-prime submodules, also generalization of quasi-prime submodules and 

approximaitly prime submodules. Various basic properties of an approximaitly semi-prime 

submodules are discussed, where a proper submodule 𝐿 of an 𝑅-module 𝑇 is called an 

approximaitly semi-prime submodule of  , if whenever 𝑎𝑛𝑡 ∈ 𝐿, where 𝑎 ∈ 𝑅, 𝑡 ∈ 𝑇 and 

𝑛 ∈ 𝑍+, implies that 𝑎𝑡 ∈ 𝐿 + 𝑠𝑜𝑐(𝑇). Furthermore the behaviors of approximaitly semi-

prime submodule in some classes of modules are studied. On the other hand several 

characterizations of this concept are introduced. 
 

Keywords: Prime submodules, Semi-prime submodules, Quasi-prime submodules, 

Approximaitly prime submodules, Approximaitly semi-prime submodules, Multiplication 

module, Socle of modules. 
 

1. Introduction 

       Throughout this article, all rings are commutative rings with identity and all modules are 

unitary. Prime submodules play an important role in module theory over commutative ring 

with identity, where a proper submodule 𝐿 of an 𝑅- module 𝑇 is called a prime, if whenever 

𝑎𝑡 ∈ 𝐿, with 𝑎 ∈ 𝑅,𝑡 ∈ 𝑇, implies that either 𝑡 ∈ 𝐿 or 𝑎 ∈ [𝐿:𝑅 𝑇] where [𝐿:𝑅 𝑇] = {𝑟 ∈

𝑅:𝑟𝑇 ⊆ 𝐿}[1]. Recently several generalization of the concept of prime submodules are 

studied in [2-5]. The concept semi-prime submodule which was first introduced in [6]. and 

extensively studied in [7]. is given as a proper submodule 𝐿 of an 𝑅-module 𝑇 is called semi-

prime submodule, if whenever𝑎𝑛𝑡 ∈ 𝐿, where 𝑎 ∈ 𝑅, 𝑡 ∈ 𝑇 and 𝑛 ∈ 𝑍+, implies that 𝑎𝑡 ∈ 𝐿. 

[7]. characterized semi-prime submodules as follows: A proper submodule 𝐿 of an 𝑅-module 

𝑇 is semi-prime if and only if whenever 𝑎2𝑡 ∈ 𝐿, where 𝑎 ∈ 𝑅, 𝑡 ∈ 𝑇, implies that 𝑎𝑡 ∈ 𝐿. 

The concept quasi-prime submodule which introduced and studied in [8]. is a strong form of a 

semi-prime submodule, where a proper submodule 𝐿 of an 𝑅- module 𝑇 is called a quasi-

prime, if whenever 𝑎𝑏𝑡 ∈ 𝐿, with 𝑎,𝑏 ∈ 𝑅,𝑡 ∈ 𝑇, implies that either 𝑎𝑡 ∈ 𝐿 or 𝑏𝑡 ∈ 𝐿. 

Ibn Al Haitham Journal for Pure and Applied Science 

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

 

Doi:10.30526/32.3.2288  

Approximaitly Semi-Prime Submodules and Some Related Concepts 

 

Department of Mathematics, College of Computer Science and Mathematics, 

University of Tikrit, Iraq. 

 Ali.shebl@st.tu.edu.iq                                                   dr.mohammadali2013@gmail.com 

Article history: Received 19 February 2019, Accepted 4 March 2019,Publish 

September 2019. 

 

mailto:Ali.shebl@st.tu.edu.iq
mailto:dr.mohammadali2013@gmail.com


  

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Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (3) 2019 

 

Recently extensive research has been done on generalizations of semi-prime submodules see 

for example [9, 10]. The socle of an 𝑅-module 𝑇 (for short 𝑠𝑜𝑐(𝑇)) is defined by the 

intersection of all essential submodules of 𝑇 [11]. where a non-zero submodule 𝑁 of an 𝑅-

module 𝑇 is called essential if 𝑁 ∩ 𝐾 ≠ (0) for all non-zero submodule 𝐾 of 𝑇 [12]. Recall 

that an ideal 𝐼 of a ring 𝑅 is a semi-prime ideal of 𝑅 if 𝑎2 ∈ 𝐼, implies that 𝑎 ∈ 𝐼. Equivalent 

𝐼 = √𝐼 = {𝑎 ∈ 𝑅: 𝑎𝑛 ∈ 𝐼 for some 𝑛 ∈ 𝑍+} [7]. Recall that a proper submodule 𝐿 of an 𝑅-

module 𝑇 is called an approximaitly prime submodule of 𝑇, if whenever 𝑎𝑡 ∈ 𝐿, where 𝑎 ∈ 𝑅, 

𝑡 ∈ 𝑇, implies that either 𝑡 ∈ 𝐿 + 𝑠𝑜𝑐(𝑇) or 𝑎 ∈ [𝐿 + 𝑠𝑜𝑐(𝑇):𝑇][2]. 
 

2. Approximaitly Semi-prime Submodules 

    In this section, we introduce the definition of approximaitly semi-prime submodule and 

give it is basic properties, examples and characterizations. 

 

Definition (1) 

     A proper submodule 𝐿 of an 𝑅-module 𝑇 is called an approximaitly semi-prime (for short 

app-semi-prime) submodule of 𝑇, if whenever 𝑎𝑛𝑡 ∈ 𝐿, where 𝑎 ∈ 𝑅, 𝑡 ∈ 𝑇 and 𝑛 ∈ 𝑍+, 

implies that 𝑎𝑡 ∈ 𝐿 + 𝑠𝑜𝑐(𝑇). An ideal 𝐼 of a ring 𝑅 is called an approximaitly semi-prime 

ideal of 𝑅 if  𝐼 is an approximaitly semi-prime submodule of 𝑅-module 𝑅. 
 

Remarks and Examples (2) 

1) It is clear that every semi-prime submodule of an 𝑅-module 𝑇 is an app-semi-prime 

submodule while the convers is not true in general as the following example shows that.  

Consider the 𝑍-module 𝑍12 and 𝐿 = 〈0̅〉 be a submodule of  𝑍12 . 

𝑠𝑜𝑐(𝑍12) = {0̅, 2̅, 4̅, 6̅, 8̅,10̅̅̅̅ }. 𝐿 is not semi-prime in 𝑍12 because 2
2. 3̅ ∈ 𝐿, but 2. 3̅ = 6 ∉ 𝐿. 

But 𝐿 is an app-semi-prime in 𝑍12 since whenever 𝑎
2𝑡̅ ∈ 𝐿, for 𝑎 ∈ 𝑅, 𝑡̅ ∈ 𝑍12 , implies that 

𝑎𝑡̅ ∈ 𝐿 + 𝑠𝑜𝑐(𝑍12) = {0̅, 2̅, 4̅, 6̅, 8̅,10̅̅̅̅ }. 

2) It is clear that every prime submodule of an 𝑅-module 𝑇 is an app-semi-prime submodule 

while the convers is not true in general as the following example shows that. 

Consider the 𝑍-module 𝑍4 and 𝐿 = 〈0̅〉 be a submodule of  𝑍4 . 𝐿 is not prime  but 𝐿 is an app-

semi-prime in 𝑍4 because  2
2. 1̅ ∈ 𝐿 but 2.1 ∉ 𝐿, while 22. 1̅ ∈ 𝐿, implies that  2. 1̅ = 2 ∈ 𝐿 +

𝑠𝑜𝑐(𝑍4) = 〈0̅〉 + {0̅, 2̅} = {0̅, 2̅}. That is for all 𝑎 ∈ 𝑍, 𝑡̅ ∈ 𝑍4 with 𝑎
2𝑡̅ ∈ 𝐿, implies that 

𝑎𝑡̅ ∈ 𝐿 + 𝑠𝑜𝑐(𝑍4). 

3) It is clear that every quasi-prime submodule of an 𝑅-module 𝑇 is an app-semi-prime 

submodule while the convers is not true in general as an example shows that.  

Consider the 𝑍-module 𝑍24 and the submodule  𝐿 = 〈6̅〉 = {0̅, 6̅,12̅̅̅̅ ,18̅̅̅̅ }. 𝐿 is not quasi-prime 

in 𝑍24 because 2.3. 1̅ ∈ 𝐿, but 2. 1̅ ∉ 𝐿 and 3. 1̅ ∉ 𝐿. But 𝐿 is an app-semi-prime in 𝑍24 since 

whenever 𝑎2𝑡̅ ∈ 𝐿 for 𝑎 ∈ 𝑅, 𝑡̅ ∈ 𝑍24 , implies that 𝑎𝑡̅ ∈ 𝐿 + 𝑠𝑜𝑐(𝑍24) = {0̅, 6̅,12̅̅̅̅ ,18̅̅̅̅ } +

{0̅, 2̅, 4̅, 6̅, 8̅,10̅̅̅̅ ,12̅̅̅̅ ,14̅̅̅̅ ,16̅̅̅̅ ,18̅̅̅̅ ,20̅̅̅̅ ,22̅̅̅̅ } = {0̅, 2̅, 4̅, 6̅, 8̅,10̅̅̅̅ ,12̅̅̅̅ ,14̅̅̅̅ ,16̅̅̅̅ ,18̅̅̅̅ ,20̅̅̅̅ ,22̅̅̅̅ }. 

4) It is clear that every approximaitly prime submodule of an 𝑅-module 𝑇 is an app-semi-

prime submodule, but the convers is not true. The following example shows that.  

The submodule 𝐿 = 6𝑍 of a 𝑍-module 𝑍  is an app-semi-prime submodule of 𝑍 (because 𝐿 is 

a semi-prime submodule of 𝑍 ), but 𝐿 is not an approximaitly prime submodule of 𝑍 because, 

if 2,3 ∈ 𝑍 such that 2.3 ∈ 6𝑍, but 3 ∉ 6𝑍 + 𝑠𝑜𝑐(𝑍) = 6𝑍 and 2 ∉ [6𝑍 + 𝑠𝑜𝑐(𝑍):𝑍] = 6𝑍, 

because 𝑠𝑜𝑐(𝑍) = (0). 

    The following results are characterizations of app-semi-prime submodules. 



  

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Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (3) 2019 

 

Proposition (3) 

     Let 𝐿 be a proper submodule of an 𝑅-module 𝑇. Then 𝐿 is an app-semi-prime submodule 

of 𝑇 if and only if 𝐽𝑛𝐸 ⊆ 𝐿 where 𝐽 is an ideal of 𝑅, 𝐸 is a submodule of 𝑇 and 𝑛 ∈ 𝑍+,  

implies that 𝐽𝐸 ⊆ 𝐿 + 𝑠𝑜𝑐(𝑇). 

Proof 

    (
        
⇒ ) Suppose that 𝐽𝑛𝐸 ⊆ 𝐿, where 𝐽 is an ideal of 𝑅, 𝐸 is a submodule of 𝑇 and 𝑛 ∈ 𝑍+. 

Now, let 𝑡 ∈ 𝐽𝐸, then 𝑡 = 𝑎1𝑡1 + 𝑎2𝑡2 + ⋯+ 𝑎𝑛𝑡𝑛, where 𝑎𝑖 ∈ 𝐽, 𝑡𝑖 ∈ 𝐸, 𝑖 = 1,2,…,𝑛, that 

is 𝑎𝑖𝑡𝑖 ∈ 𝐽𝐸 for each 𝑖 = 1,2,…,𝑛, it follows that 𝑎
𝑛
𝑖𝑡𝑖 ∈ 𝐽

𝑛𝐸 ⊆ 𝐿, that is 𝑎𝑛𝑖𝑡𝑖 ∈ 𝐿. But 𝐿 is 

an app-semi-prime submodule of 𝑇, implies that 𝑎𝑖𝑡𝑖 ∈ 𝐿 + 𝑠𝑜𝑐(𝑇) for each 𝑖 = 1,2,…,𝑛, 

hence 𝑡 ∈ 𝐿 + 𝑠𝑜𝑐(𝑇), it follows that 𝐽𝐸 ⊆ 𝐿 + 𝑠𝑜𝑐(𝑇). 

    (
         
⇐  ) Let 𝑎𝑛𝑡 ∈ 𝐿, where 𝑎 ∈ 𝑅, 𝑡 ∈ 𝑇 and 𝑛 ∈ 𝑍+, implies that 〈𝑎𝑛〉𝑅𝑡 ⊆ 𝐿, that is 

〈𝑎〉𝑛𝑅𝑡 ⊆ 𝐿. Thus by hypothesis we have〈𝑎〉𝑅𝑡 ⊆ 𝐿 + 𝑠𝑜𝑐(𝑇), implies that 𝑎𝑡 ∈ 〈𝑎〉𝑅𝑡 ⊆ 𝐿 +

𝑠𝑜𝑐(𝑇), thus 𝑎𝑡 ∈ 𝐿 + 𝑠𝑜𝑐(𝑇). Therefore  𝐿 is an app-semi-prime submodule of  𝑇. 
 

Corollary (4) 

     Let 𝐿 be a proper submodule of an 𝑅-module 𝑇. Then 𝐿 is an app-semi-prime submodule 

of 𝑇 if and only if 𝐽𝑛𝑇 ⊆ 𝐿 where 𝐽 is an ideal of 𝑅, 𝑛 ∈ 𝑍+,  implies that 𝐽𝑇 ⊆ 𝐿 + 𝑠𝑜𝑐(𝑇). 

Proof 

    It follows by Proposition (2.3). 
 

Corollary (5) 

     Let 𝐿 be a proper submodule of an 𝑅-module 𝑇. Then 𝐿 is an app-semi-prime submodule 

of 𝑇 if and only if 𝐽2𝐸 ⊆ 𝐿 where 𝐽 is an ideal of 𝑅 and 𝐸 is a submodule of 𝑇,  implies that 

𝐽𝐸 ⊆ 𝐿 + 𝑠𝑜𝑐(𝑇). 
 

Proposition (6) 

     Let 𝐿 be a proper submodule of an 𝑅-module 𝑇. Then 𝐿 + 𝑠𝑜𝑐(𝑇) is an app-semi-prime 

submodule of 𝑇 if and only if [𝐿 + 𝑠𝑜𝑐(𝑇):𝑇] is a semi-prime ideal of 𝑅 (hence an app-semi-

prime). 

Proof 

    (
        
⇒ ) Let 𝑟 ∈ √[𝐿 + 𝑠𝑜𝑐(𝑇):𝑇], implies that 𝑟𝑛 ∈ [𝐿 + 𝑠𝑜𝑐(𝑇):𝑇] for some 𝑛 ∈ 𝑍+, it 

follows that 𝑟𝑛𝑇 ⊆ 𝐿 + 𝑠𝑜𝑐(𝑇), then 𝑟𝑛𝑡 ∈ 𝐿 + 𝑠𝑜𝑐(𝑇) for all 𝑡 ∈ 𝑇. But 𝐿 + 𝑠𝑜𝑐(𝑇) is an 

app-semi-prime, implies that 𝑟𝑡 ∈ 𝐿 + 𝑠𝑜𝑐(𝑇) + 𝑠𝑜𝑐(𝑇) = 𝐿 + 𝑠𝑜𝑐(𝑇) for all 𝑡 ∈ 𝑇. That is 

𝑟𝑇 ⊆ 𝐿 + 𝑠𝑜𝑐(𝑇), implies that 𝑟 ∈ [𝐿 + 𝑠𝑜𝑐(𝑇):𝑇]. Thus √[𝐿 + 𝑠𝑜𝑐(𝑇):𝑇] ⊆ [𝐿 +

𝑠𝑜𝑐(𝑇):𝑇], but [𝐿 + 𝑠𝑜𝑐(𝑇):𝑇] ⊆ √[𝐿 + 𝑠𝑜𝑐(𝑇):𝑇], it follows that [𝐿 + 𝑠𝑜𝑐(𝑇):𝑇] =

√[𝐿 + 𝑠𝑜𝑐(𝑇):𝑇], hence [𝐿 + 𝑠𝑜𝑐(𝑇):𝑇] is a semi-prime ideal of 𝑅( hence [𝐿 + 𝑠𝑜𝑐(𝑇):𝑇] is 

an app-semi-prime). 

    (
         
⇐  ) Let  𝑎𝑛𝑡 ∈ 𝐿 + 𝑠𝑜𝑐(𝑇), where 𝑎 ∈ 𝑅, 𝑡 ∈ 𝑇 and 𝑛 ∈ 𝑍+, implies that 𝑎𝑛𝑇 ⊆ 𝐿 +

𝑠𝑜𝑐(𝑇), that is 𝑎𝑛 ∈ [𝐿 + 𝑠𝑜𝑐(𝑇):𝑇]. Since [𝐿 + 𝑠𝑜𝑐(𝑇):𝑇] is a semi-prime ideal of 𝑅 then 

𝑎 ∈ [𝐿 + 𝑠𝑜𝑐(𝑇):𝑇], it follows that 𝑎𝑇 ⊆ 𝐿 + 𝑠𝑜𝑐(𝑇), implies that 𝑎𝑡 ∈ 𝐿 + 𝑠𝑜𝑐(𝑇) for all 

𝑡 ∈ 𝑇. Therefore 𝐿 + 𝑠𝑜𝑐(𝑇) is an app-semi-prime submodule of 𝑇. 
 

Proposition (7) 

     Let 𝐿 be a proper submodule of an 𝑅-module 𝑇 with 𝑠𝑜𝑐(𝑇) ⊆ 𝐿. Then 𝐿 is an app-semi-

prime submodule of 𝑇 if and only if  [𝐿:𝑇 𝑎
𝑛] = [𝐿:𝑇 𝑎] for 𝑎 ∈ 𝑅 and some 𝑛 ∈ 𝑍

+. 



  

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Proof 

    (
        
⇒ ) Let 𝑡 ∈ [𝐿:𝑇 𝑎

𝑛], implies that 𝑎𝑛𝑡 ∈ 𝐿. But 𝐿 is an app-semi-prime submodule of 𝑇, 

then 𝑎𝑡 ∈ 𝐿 + 𝑠𝑜𝑐(𝑇). But 𝑠𝑜𝑐(𝑇) ⊆ 𝐿, implies that 𝐿 + 𝑠𝑜𝑐(𝑇) = 𝐿, thus 𝑎𝑡 ∈ 𝐿, it follows 

that 𝑡 ∈ [𝐿:𝑇 𝑎], hence [𝐿:𝑇 𝑎
𝑛] ⊆ [𝐿:𝑇 𝑎]. But [𝐿:𝑇 𝑎] ⊆ [𝐿:𝑇 𝑎

𝑛], so we get [𝐿:𝑇 𝑎
𝑛] =

[𝐿:𝑇 𝑎]. 

    (
         
⇐  ) Suppose that 𝑎𝑛𝑡 ∈ 𝐿, where 𝑎 ∈ 𝑅, 𝑡 ∈ 𝑇 and 𝑛 ∈ 𝑍+, it follows that 𝑡 ∈ [𝐿:𝑇 𝑎

𝑛] =

[𝐿:𝑇 𝑎], implies that 𝑡 ∈ [𝐿:𝑇 𝑎], so 𝑎𝑡 ∈ 𝐿 ⊆ 𝐿 + 𝑠𝑜𝑐(𝑇). That is 𝑎𝑡 ∈ 𝐿 + 𝑠𝑜𝑐(𝑇), hence 𝐿 

is an app-semi-prime submodule of 𝑇. 

We recall the following Lemmas before we introduce the next results. 
 

Lemma (8) [11, Lemma 2.3.15]. 

       Let 𝑇 be 𝑅-module and 𝐿,𝐾,𝐸 are submodules of 𝑇 with 𝐾 ⊆ 𝐸, then (𝐿 + 𝐾) ∩ 𝐸 =

(𝐿 ∩ 𝐸) + (𝐾 ∩ 𝐸) = (𝐿 ∩ 𝐸) + 𝐾. 

Lemma (9) [13, Cor. 9.9]. 

       Let 𝑇 be 𝑅-module and 𝐿 be a submodule of  𝑇, then 𝑠𝑜𝑐(𝐿) = 𝐿 ∩ 𝑠𝑜𝑐(𝑇). 
 

Proposition (10) 

     Let 𝐿 and 𝐸 are proper submodules of an 𝑅-module 𝑇 withe 𝐿 ⊊ 𝐸 and 𝐿 is an app-semi-

prime submodule of 𝑇. Then 𝐿 is an app-semi-prime submodule of 𝐸. 

Proof  

     Suppose that 𝑎𝑛𝑡 ∈ 𝐿, where 𝑎 ∈ 𝑅, 𝑡 ∈ 𝐸 ⊊ 𝑇 and 𝑛 ∈ 𝑍+. But 𝐿 is an app-semi-prime 

submodule of 𝑇, implies that 𝑎𝑡 ∈ 𝐿 + 𝑠𝑜𝑐(𝑇), but 𝑡 ∈ 𝐸, implies that 𝑎𝑡 ∈ 𝐸, then 𝑎𝑡 ∈

(𝐿 + 𝑠𝑜𝑐(𝑇)) ∩ 𝐸, thus by Lemma(2.8) we have 𝑎𝑡 ∈ (𝐿 ∩ 𝐸) + (𝑠𝑜𝑐(𝑇) ∩ 𝐸) ⊆ 𝐿 +

𝑠𝑜𝑐(𝑇) ∩ 𝐸. So by Lemma (2.9) we have 𝑎𝑡 ∈ 𝐿 + 𝑠𝑜𝑐(𝐸). Thus  𝐿 is an app-semi-prime 

submodule of  𝐸. 
 

Proposition (11) 

      Let 𝐿 and 𝐸 are proper submodules of an 𝑅-module 𝑇 withe 𝐿 ⊈ 𝐸 and  𝑠𝑜𝑐(𝑇) ⊆ 𝐸. If  𝐿 

is an app-semi-prime submodule of 𝑇. Then 𝐿 ∩ 𝐸 is an app-semi-prime submodule of 𝐸. 

Proof 

    Since 𝐿 ⊈ 𝐸 then 𝐿 ∩ 𝐸 is a proper in 𝐸. Suppose that 𝑎𝑛𝑡 ∈ 𝐿 ∩ 𝐸, where 𝑎 ∈ 𝑅, 𝑡 ∈ 𝐸 ⊊

𝑇 and 𝑛 ∈ 𝑍+, implies that 𝑎𝑛𝑡 ∈ 𝐿. But 𝐿 is an app-semi-prime submodule of 𝑇, then 

𝑎𝑡 ∈ 𝐿 + 𝑠𝑜𝑐(𝑇), it follows that 𝑎𝑡 ∈ (𝐿 + 𝑠𝑜𝑐(𝑇)) ∩ 𝐸, so by Lemma (2.8) we have 

𝑎𝑡 ∈ (𝐿 ∩ 𝐸) + (𝐸 ∩ 𝑠𝑜𝑐(𝑇)). But by lemma (2.9) we have  𝐸 ∩ 𝑠𝑜𝑐(𝑇) = 𝑠𝑜𝑐(𝐸). Hence 

𝑎𝑡 ∈ (𝐿 ∩ 𝐸) + 𝑠𝑜𝑐(𝐸). Thus  𝐿 ∩ 𝐸 is an app-semi-prime submodule of 𝐸. 
 

Remark (12) 

    If 𝐿 and 𝐸 are two app-semi-prime submodules of an 𝑅-module 𝑇, then 𝐿 ∩ 𝐸 is not 

necessary an app-semi-prime submodule of 𝑇 as the following example shows that. 

    Let = 𝑍⨁𝑍4 , 𝑅 = 𝑍, 𝐿 = 𝑍(1, 0̅), 𝐸 = 𝑍(1, 1̅), where 𝐿 and 𝐸 are app-semi-prime 

submodules of 𝑇. Then 𝐿 ∩ 𝐸 = {(0,0̅),(4, 0̅),(8, 0̅),……} is not an app-semi-prime 

submodule of 𝑇, since 22(1, 0̅) ∈ 𝐿 ∩ 𝐸, but 2(1, 0̅) ∉ 𝐿 ∩ 𝐸 + 𝑠𝑜𝑐(𝑇) where 𝑠𝑜𝑐(𝑇) = (0). 
 

Proposition (13) 

     Let 𝐿 and 𝐸 are two app-semi-prime submodules of an 𝑅-module 𝑇 with 𝑠𝑜𝑐(𝑇) ⊆ 𝐿 or 

𝑠𝑜𝑐(𝑇) ⊆ 𝐸. Then 𝐿 ∩ 𝐸 is an app-semi-prime submodule of 𝑇. 



  

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Proof 

     Suppose that 𝑎𝑛𝑡 ∈ 𝐿 ∩ 𝐸, where 𝑎 ∈ 𝑅, 𝑡 ∈ 𝑇 and 𝑛 ∈ 𝑍+, implies that 𝑎𝑛𝑡 ∈ 𝐿 and 

𝑎𝑛𝑡 ∈ 𝐿. But 𝐿 and 𝐸 are  app-semi-prime submodules of 𝑇, then 𝑎𝑡 ∈ 𝐿 + 𝑠𝑜𝑐(𝑇) and 

𝑎𝑡 ∈ 𝐸 + 𝑠𝑜𝑐(𝑇), it follows that 𝑎𝑡 ∈ (𝐿 + 𝑠𝑜𝑐(𝑇)) ∩ (𝐸 + 𝑠𝑜𝑐(𝑇)). If 𝑠𝑜𝑐(𝑇) ⊆ 𝐸, then 

𝑎𝑡 ∈ (𝐿 + 𝑠𝑜𝑐(𝑇)) ∩ 𝐸 and by Lemma (2.8) we have 𝑎𝑡 ∈ (𝐿 ∩ 𝐸) + 𝑠𝑜𝑐(𝑇). Similarly if 

𝑠𝑜𝑐(𝑇) ⊆ 𝐿, we get 𝑎𝑡 ∈ (𝐿 ∩ 𝐸) + 𝑠𝑜𝑐(𝑇). Hence 𝐿 ∩ 𝐸 is an app-semi-prime submodule of 

𝑇. 
 

Remark (14) 

     If 𝐿 and 𝐸 are submodules of an 𝑅-module 𝑇 with 𝐿 ⊆ 𝐸, and 𝐸 ia an app-semi-prime 

submodule of 𝑇, then 𝐿 is not an app-semi-prime submodule of 𝑇, the following example 

shows that. 

     Let 𝐿 = 12𝑍 and 𝐸 = 6𝑍 are submodules of a 𝑍-module 𝑍, 𝐿 ⊆ 𝐸 and 𝐸 is an app-semi-

prime submodule of Z, but 𝐿 = 12𝑍 is not an app-semi-prime submodule because 22.3 ∈

12𝑍, but 2.3 ∉ 12𝑍 + 𝑠𝑜𝑐(𝑍). 

      Recall that an 𝑅-module 𝑇 is multiplication if every submodule 𝐿 of 𝑇 is of the form 

𝐿 = 𝐼𝑇 for some ideal 𝐼 of 𝑅. In particular 𝐿 = [𝐿:𝑅 𝑇]𝑇  [14]. Recall that for any submodules  

𝐾 and 𝐹 of a multiplication 𝑅-module 𝑇 with 𝐾 = 𝐼𝑇 and 𝐹 = 𝐽𝑇 for some ideals 𝐼 and 𝐽 of 

𝑅. The product 𝐾𝐹 = 𝐼𝑇.𝐽𝑇 = 𝐼𝐽𝑇 that is 𝐾𝐹 = 𝐼𝐹. In particular 𝐾𝑇 = 𝐼𝑇𝑇 = 𝐼𝑇 = 𝐾. Also 

for any 𝑡 ∈ 𝑇, we have 𝐾𝑡 = 𝐾〈𝑡〉 = 𝐼𝑡 [15]. 

      The following propositions are characterizations of app-semi-prime submodules in the 

class of multiplication modules. 
 

Proposition (15) 

     Let 𝐿 be a proper submodule of a multiplication 𝑅-module 𝑇. Then 𝐿 is an app-semi-prime 

submodule of 𝑇 if and only if 𝐾𝑛𝐹 ⊆ 𝐿 implies that 𝐾𝐹 ⊆ 𝐿 + 𝑠𝑜𝑐(𝑇), where 𝐾,𝐹 are 

submodules of  𝑇, 𝑛 ∈ 𝑍+. 

Proof 

    (
        
⇒ ) Suppose that 𝐾𝑛𝐹 ⊆ 𝐿, where 𝐾,𝐹 are submodules of 𝑇, 𝑛 ∈ 𝑍+. Since 𝑇 is a 

multiplication then 𝐾 = 𝐼𝑇 and 𝐹 = 𝐽𝑇 for some ideals 𝐼, 𝐽 of 𝑅. Thus 𝐾𝑛𝐹 = (𝐼𝑇)𝑛𝐽𝑇 =

𝐼𝑛(𝐽𝑇) ⊆ 𝐿. But 𝐿 is an app-semi-prime, then by Proposition (2.3) we have  𝐼(𝐽𝑇) ⊆ 𝐿 +

𝑠𝑜𝑐(𝑇). That is 𝐼𝐹 ⊆ 𝐿 + 𝑠𝑜𝑐(𝑇), so 𝐾𝐹 ⊆ 𝐿 + 𝑠𝑜𝑐(𝑇). 

    (
         
⇐  ) Suppose that   𝐼𝑛𝐹 ⊆ 𝐿, where 𝐼 is an ideal of 𝑅, 𝐹 is a submodule of 𝑇 and 𝑛 ∈ 𝑍+. 

Since 𝑇 is a multiplication then 𝐹 = 𝐽𝑇 for some ideal 𝐽 of 𝑅. That is 𝐼𝑛𝐽𝑇 ⊆ 𝐿 implies that 

𝐾𝑛𝐹 ⊆ 𝐿 so by hypothesis 𝐾𝐹 ⊆ 𝐿 + 𝑠𝑜𝑐(𝑇), thus 𝐼𝐹 ⊆ 𝐿 + 𝑠𝑜𝑐(𝑇). Hence by Proposition 

(2.3)  𝐿 is an app-semi-prime submodule of 𝑇. 
 

Proposition (16) 

     Let 𝐿 be a proper submodule of a multiplication 𝑅-module 𝑇. Then the following 

statements are equivalent: 

1)  𝐿 is an app-semi-prime submodule of  𝑇. 

2)  𝑡𝑛 ∈ 𝐿 implies that 𝑡 ∈ 𝐿 + 𝑠𝑜𝑐(𝑇) for every 𝑡 ∈ 𝑇. 

3)  √𝐿 ⊆ 𝐿 + 𝑠𝑜𝑐(𝑇). 

4) 𝐹1𝐹2 ……𝐹𝑗 ⊆ 𝐿, implies that 𝐹1 ∩ 𝐹2 ∩ ……∩ 𝐹𝑗 ⊆ 𝐿 + 𝑠𝑜𝑐(𝑇) for every submodules 

𝐹1,𝐹2,……,𝐹𝑗 of 𝑇 and 𝑗 ∈ 𝑍
+. 



  

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Proof 

      (1) 
        
⇒ (2) Let 𝑡𝑛 ∈ 𝐿 where 𝑡 ∈ 𝑇 and for some 𝑛 ∈ 𝑍+, then 〈𝑡𝑛〉 ⊆ 𝐿. But 𝑇 is a 

multiplication 𝑅-module, then 〈𝑡〉 = 𝐼𝑇 for some ideal 𝐼 of 𝑅, so 〈𝑡𝑛〉 = 𝐼𝑛𝑇 ⊆ 𝐿. Since 𝐿 is 

an app-semi-prime submodule of  𝑇, then by Corollary (2.4) we have 𝐼𝑇 ⊆ 𝐿 + 𝑠𝑜𝑐(𝑇). That 

is 〈𝑡〉 ⊆ 𝐿 + 𝑠𝑜𝑐(𝑇), implies that 𝑡 ⊆ 𝐿 + 𝑠𝑜𝑐(𝑇). 

 

 (2) 
        
⇒ (3) Let  𝑡 ∈ √𝐿 , implies that 𝑡𝑛 ∈ 𝐿 for some 𝑛 ∈ 𝑍+, so by hypothesis 𝑡 ∈ 𝐿 +

𝑠𝑜𝑐(𝑇). Thus √𝐿 ⊆ 𝐿 + 𝑠𝑜𝑐(𝑇). 

(3) 
        
⇒ (4) Suppose that 𝐹1𝐹2 ……𝐹𝑗 ⊆ 𝐿 where  𝐹1,𝐹2,……,𝐹𝑗  are submodules of 𝑇 and 

𝑗 ∈ 𝑍+. Let 𝑡 ∈ 𝐹1 ∩ 𝐹2 ∩ ……∩ 𝐹𝑗 then 𝑡 ∈ 𝐹𝑖 for each = 1,2,……,𝑗 , so 𝑡
𝑗 ∈ 𝐹1𝐹2 ……𝐹𝑗 ⊆

𝐿, it follows that 𝑡𝑗 ∈ 𝐿, so 𝑡 ∈ √𝐿. But by hypothesis √𝐿 ⊆ 𝐿 + 𝑠𝑜𝑐(𝑇), then 𝑡 ∈ 𝐿 +

𝑠𝑜𝑐(𝑇). Thus 𝐹1 ∩ 𝐹2 ∩ ……∩ 𝐹𝑗 ⊆ 𝐿 + 𝑠𝑜𝑐(𝑇). 

(3) 
        
⇒ (4) Let 𝐼𝑛𝐸 ⊆ 𝐿, where 𝐼 is an ideal of 𝑅, 𝐸 is a submodule of 𝑇, and 𝑛 ∈ 𝑍+. That is 

(𝐼𝑇)(𝐼𝑇)……(𝐼𝑇) ⊆ 𝐿, so by hypothesis (𝐼𝑇) ∩ (𝐼𝑇)……∩ (𝐼𝑇) ⊆ 𝐿 + 𝑠𝑜𝑐(𝑇). Implies that 

𝐼𝑇 ⊆ 𝐿 + 𝑠𝑜𝑐(𝑇). Thus by Corollary (2.4) 𝐿 is an app-semi-prime submodule of  𝑇. 
 

Remark (17) 

       If 𝐿 is an app-semi-prime submodule of an 𝑅-module 𝑇, then [𝐿:𝑅 𝑇] is not necessary 

app-semi-prime ideal of 𝑅. The following example shows that. 

Consider the 𝑍-module 𝑍8  and a submodule 𝐿 = 〈0̅〉. 𝐿 is an app-semi-prime submodule 

since 𝑠𝑜𝑐(𝑍8) = 〈2̅〉 = {0̅, 2̅, 4̅, 6̅} and  2
2. 2̅ ∈ 𝐿, implies that 2. 2̅ = 4 ∈ 𝐿 + 𝑠𝑜𝑐(𝑍8) =

{0̅, 2̅, 4̅, 6̅}, where 2 ∈ 𝑍,2̅ ∈ 𝑍8 . So for all 𝑎 ∈ 𝑅, 𝑡̅ ∈ 𝑍8 , such that 𝑎
𝑛𝑡̅ ∈ 𝐿 for some 

𝑛 ∈ 𝑍+,  implies that 𝑎𝑡̅ ∈ 𝐿 + 𝑠𝑜𝑐(𝑍8). But [𝐿:𝑍 𝑇] = [〈0̅〉:𝑍 𝑍8] = 8𝑍 is not app-semi-prime 

ideal in 𝑍, since 22.2 ∈ 8𝑍 but 2.2 ∉ 8𝑍 + 𝑠𝑜𝑐(𝑍) = 8𝑍 + (0) = 8𝑍. 
 

Proposition (18) 

       Let 𝐿 be an app-semi-prime submodule of an 𝑅-module 𝑇, with 𝑠𝑜𝑐(𝑇) ⊆ 𝐿. Then 

[𝐿:𝑅 𝑇] is an app-semi-prime ideal of 𝑅. 

Proof 

       Suppose that 𝑎𝑛𝑠 ∈ [𝐿:𝑅 𝑇], where 𝑎,𝑠 ∈ 𝑅, 𝑛 ∈ 𝑍
+, then 𝑎𝑛𝑠 𝑇 ⊆ 𝐿. That is  𝑎𝑛(𝑠 𝑇) ⊆

𝐿, implies that 𝑎𝑛(𝑠 𝑡) ∈ 𝐿 for all 𝑡 ∈ 𝑇. But 𝐿 is an app-semi-prime submodule of 𝑇, implies 

that 𝑎(𝑠 𝑡) ∈ 𝐿 + 𝑠𝑜𝑐(𝑇), that is 𝑎𝑠 𝑇 ⊆ 𝐿 + 𝑠𝑜𝑐(𝑇). But 𝑠𝑜𝑐(𝑇) ⊆ 𝐿, implies that 𝐿 +

𝑠𝑜𝑐(𝑇) = 𝐿, thus 𝑎𝑠 𝑇 ⊆ 𝐿, it follows that 𝑎𝑠 ∈ [𝐿:𝑅 𝑇] ⊆ [𝐿:𝑅 𝑇] + 𝑠𝑜𝑐(𝑅). Therefore 

[𝐿:𝑅 𝑇] is an app-semi-prime ideal of 𝑅. 

  We need to introduce the following Lemma which appear in [14]. 
 

Lemma (19)[14, Coro. 2.14]. 

     Let 𝑇 be a faithful multiplication 𝑅-module then 𝑠𝑜𝑐(𝑅)𝑇 = 𝑠𝑜𝑐(𝑇). 

 

Proposition (20) 

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

an app-semi-prime submodule of 𝑇 if and only if [𝐿:𝑅 𝑇] is an app-semi-prime ideal of 𝑅. 

 

 



  

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Proof 

    (
        
⇒ ) Suppose that 𝐿 is an app-semi-prime submodule of 𝑇, to prove that √[𝐿:𝑅 𝑇] ⊆

[𝐿:𝑅 𝑇] + 𝑠𝑜𝑐(𝑅) by proposition (2.16), we get [𝐿:𝑅 𝑇] is an app-semi-prime ideal of 𝑅. Let 

𝑎 ∈ √[𝐿:𝑅 𝑇], implies that 𝑎
𝑛 ∈ [𝐿:𝑅 𝑇] for some 𝑛 ∈ 𝑍

+, it follows that 𝑎𝑛𝑇 ⊆ 𝐿, that is 

𝑎𝑛𝑡 ∈ 𝐿 for all 𝑡 ∈ 𝑇. But 𝐿 is an app-semi-prime submodule of 𝑇, then 𝑎𝑡 ∈ 𝐿 + 𝑠𝑜𝑐(𝑇) for 

all 𝑡 ∈ 𝑇. That is 𝑎𝑇 ⊆ 𝐿 + 𝑠𝑜𝑐(𝑇). Since 𝑇 is a multiplication𝑅-module, so 𝐿 = [𝐿:𝑅 𝑇]𝑇, 

and since 𝑇 is faithful multiplication, so by Lemma (2.19) 𝑠𝑜𝑐(𝑇) = 𝑠𝑜𝑐(𝑅)𝑇. thus 𝑎𝑇 ⊆

[𝐿:𝑅 𝑇]𝑇 + 𝑠𝑜𝑐(𝑅)𝑇, it follows that 𝑎 ∈ [𝐿:𝑅 𝑇] + 𝑠𝑜𝑐(𝑅). Hence √[𝐿:𝑅 𝑇] ⊆ [𝐿:𝑅 𝑇] +

𝑠𝑜𝑐(𝑅), therefore [𝐿:𝑅 𝑇]  is an app-semi-prime ideal of 𝑅. 

    (
         
⇐  ) Suppose that [𝐿:𝑅 𝑇] is an app-semi-prime ideal of 𝑅, and let  𝑎

𝑛𝑡 ∈ 𝐿, where 𝑎 ∈ 𝑅, 

𝑡 ∈ 𝑇 and 𝑛 ∈ 𝑍+, it follows that  𝑎𝑛𝑇 ⊆ 𝐿, implies that  𝑎𝑛 ∈ [𝐿:𝑅 𝑇] so 𝑎 ∈ √[𝐿:𝑅 𝑇]. Since 

[𝐿:𝑅 𝑇] is an app-semi-prime ideal of 𝑅 then by proposition (2.16) √[𝐿:𝑅 𝑇] ⊆ [𝐿:𝑅 𝑇] +

𝑠𝑜𝑐(𝑅), it follows that 𝑎 ∈ [𝐿:𝑅 𝑇] + 𝑠𝑜𝑐(𝑅), so 𝑎𝑇 ⊆ [𝐿:𝑅 𝑇]𝑇 + 𝑠𝑜𝑐(𝑅)𝑇. Since 𝑇 is 

faithful multiplication, then by lemma (2.19) we get 𝑎𝑇 ⊆ 𝐿 + 𝑠𝑜𝑐(𝑇), hence 𝑎𝑡 ∈ 𝐿 +

𝑠𝑜𝑐(𝑇) for all 𝑡 ∈ 𝑇. Thus 𝐿 is an app-semi-prime submodule of 𝑇. 

        Recall that an 𝑅-module 𝑇 is a non-singular provided that 𝑍(𝑇) = 𝑇, where 𝑍(𝑇) = {𝑥 ∈

𝑇 ∶ 𝑥𝐼 = 0 for some essential ideal 𝐼 of 𝑅} [12]. 

    We need the following Lemma which appears in [12]. before we introduced the next result. 
 

Lemma (21) [12, Coro. 1.26].  

      If  𝑇 is a non-singular 𝑅-module, then 𝑠𝑜𝑐(𝑅)𝑇 = 𝑠𝑜𝑐(𝑇). 
 

Proposition (22) 

      Let 𝐿 be a proper submodule of non-singular multiplication 𝑅-module 𝑇. Then 𝐿 is an 

app-semi-prime submodule of 𝑇 if and only if [𝐿:𝑅 𝑇] is an app-semi-prime ideal of 𝑅. 

Proof 

      Follows by similar steps of proposition (2.20) and used lemma (2.21), and proposition 

(2.16). 

     We need the following Lemma which appear in [16]. before we introduce the next result. 
 

Lemma (23)[16, Coro. Of Theo. 9]. 

        Let 𝑇 be a finitely generated multiplication 𝑅-module and 𝐼, 𝐽 are ideals of a ring 𝑅. 

Then 𝐼𝑇 ⊆ 𝐽𝑇 if and only if 𝐼 ⊆ 𝐽 + 𝑎𝑛𝑛(𝑇). 
 

Proposition (24) 

        Let 𝑇 be a faithful finitely generated multiplication 𝑅-module and 𝐽 be an app-semi-

prime ideal of 𝑅. Then 𝐽𝑇 is an app-semi-prime submodule of 𝑇.  

Proof 

        Let  𝑎𝑛𝐸 ⊆ 𝐽𝑇, where 𝑎 ∈ 𝑅 , 𝐸 be a submodule of 𝑇 and 𝑛 ∈ 𝑍+. Since 𝑇 is a 

multiplication, then 𝐸 = 𝐼𝑇 for some ideal 𝐼 of 𝑅. That is 𝑎𝑛𝐼𝑇 ⊆ 𝐽𝑇. But 𝑇 is a finitely 

generated, so by Lemma (2.23) we have 𝑎𝑛𝐼 ⊆ 𝐽 + 𝑎𝑛𝑛(𝑇), but 𝑇 is faithful, then 𝑎𝑛𝑛(𝑇) =

(0), hence 𝑎𝑛𝐼 ⊆ 𝐽, but 𝐽 an app-semi-prime ideal of 𝑅, then by proposition (2.3) 𝑎𝐼 ⊆ 𝐽 +

𝑠𝑜𝑐(𝑅). That is 𝑎𝐼𝑇 ⊆ 𝐽𝑇 + 𝑠𝑜𝑐(𝑅)𝑇, then by Lemma (2.19) we get 𝑎𝐸 ⊆ 𝐽𝑇 + 𝑠𝑜𝑐(𝑇). 

Hence 𝐽𝑇 is an app-semi-prime submodule of 𝑇. 
 



  

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Proposition (25) 

        Let 𝑇 be a finitely generated multiplication non-singular 𝑅-module and 𝐽 be an app-semi-

prime ideal of 𝑅 with 𝑎𝑛𝑛(𝑇) ⊆ 𝐽. Then 𝐽𝑇 is an app-semi-prime submodule of 𝑇.  

Proof 

     Suppose that  𝑎𝑛𝐿 ⊆ 𝐽𝑇, where 𝑎 ∈ 𝑅 , 𝐿 be a submodule of 𝑇 and 𝑛 ∈ 𝑍+. Since 𝑇 is a 

multiplication, then 𝐿 = 𝐼𝑇 for some ideal 𝐼 of 𝑅. That is 𝑎𝑛𝐼𝑇 ⊆ 𝐽𝑇. But 𝑇 is a finitely 

generated, so by Lemma (2.23) we have 𝑎𝑛𝐼 ⊆ 𝐽 + 𝑎𝑛𝑛(𝑇), but 𝑎𝑛𝑛(𝑇) ⊆ 𝐽, implies that 

𝐽 + 𝑎𝑛𝑛(𝑇) = 𝐽, so 𝑎𝑛𝐼 ⊆ 𝐽, but 𝐽 an app-semi-prime ideal of 𝑅, we have 𝑎𝐼 ⊆ 𝐽 + 𝑠𝑜𝑐(𝑅). 

That is 𝑎𝐼𝑇 ⊆ 𝐽𝑇 + 𝑠𝑜𝑐(𝑅)𝑇, then by Lemma (2.19) we get 𝑎𝐿 ⊆ 𝐽𝑇 + 𝑠𝑜𝑐(𝑇). Hence 𝐽𝑇 is 

an app-semi-prime submodule of 𝑇. 
 

Theorem (26) 

        Let 𝑇 be a faithful finitely generated multiplication 𝑅-module and 𝐿 be a proper 

submodule of 𝑇. Then the following statements are equivalent. 

1) 𝐿 is an app-semi-prime submodule of 𝑇. 

2) [𝐿:𝑅 𝑇] is an app-semi-prime ideal of 𝑅. 

3) 𝐿 = 𝐼𝑇 for some app-semi-prime ideal 𝐼 of 𝑅. 

Proof 

(1) 
          
⇐  (2) Follows by proposition (2.20). 

(2) 
        
⇒  (3) Since [𝐿:𝑅 𝑇] is an app-semi-prime ideal of 𝑅, and 𝐿 = [𝐿:𝑅 𝑇]𝑇 for 𝑇 is a 

multiplication, implies that 𝐿 = 𝐼𝑇 where [𝐿:𝑅 𝑇] is an app-semi-prime ideal of 𝑅. 

 (3) 
        
⇒  (2) Suppose that 𝐿 = 𝐼𝑇 for some app-semi-prime ideal 𝐼 of 𝑅. But 𝑇 is a faithful 

finitely generated multiplication, then by Lemma (2.23) 𝐼 = [𝐿:𝑅 𝑇], hence [𝐿:𝑅 𝑇] is an app-

semi-prime ideal of 𝑅. 
 

Theorem (27) 

       Let 𝑇 be non-singular finitely generated multiplication 𝑅-module and 𝐿 be a proper 

submodule of 𝑇. Then the following statements are equivalent. 

1) 𝐿 is an app-semi-prime submodule of 𝑇. 

2) [𝐿:𝑅 𝑇] is an app-semi-prime ideal of 𝑅. 

3) 𝐿 = 𝐼𝑇 for some app-semi-prime ideal 𝐼 of 𝑅 with 𝑎𝑛𝑛(𝑇) ⊆ 𝐼. 

Proof 

(1) 
          
⇐  (2) Follows by proposition (2.22). 

(2) 
        
⇒  (3) Since [𝐿:𝑅 𝑇] is an app-semi-prime ideal of 𝑅, and 𝐿 = [𝐿:𝑅 𝑇]𝑇 for 𝑇 is a 

multiplication, then 𝐿 = 𝐼𝑇 and 𝐼 = [𝐿:𝑅 𝑇] is an app-semi-prime ideal of 𝑅 with [(0):𝑅 𝑇] =

𝑎𝑛𝑛(𝑇) ⊆ [𝐿:𝑅 𝑇]. 

 (3) 
        
⇒  (2) Suppose that 𝐿 = 𝐼𝑇 for some app-semi-prime ideal 𝐼 of 𝑅 with 𝑎𝑛𝑛(𝑇) ⊆ 𝐼. But 

𝑇 is a multiplication, we have 𝐼 = [𝐿:𝑅 𝑇]𝑇 = 𝐼𝑇, that is [𝐿:𝑅 𝑇] = 𝐼 + 𝑎𝑛𝑛(𝑇) = 𝐼, it follows 

that  [𝐿:𝑅 𝑇] is an app-semi-prime ideal of 𝑅. 

        Recall that an envelope of a submodule 𝐿 of an 𝑅-module denoted by 𝐸𝑇(𝐿) defined by 

𝐸𝑇(𝐿) = {𝑎𝑡:𝑎 ∈ 𝑅,𝑡 ∈ 𝑇 such that 𝑎
𝑛𝑡 ∈ 𝐿,𝑛 ∈ 𝑍+} and 𝐿 ⊆ 𝐸𝑇(𝐿) [17]. 

 

 

 



  

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Proposition (28) 

      Let 𝐿 be a proper submodule of an 𝑅-module 𝑇. Then 𝐿 is an app-semi-prime submodule 

of 𝑇 if and only if 𝐸𝑇(𝐿) ⊆ 𝐿 + 𝑠𝑜𝑐(𝑇). 

Proof 

      (
        
⇒ ) Let ∈ 𝐸𝑇(𝐿) , implies that 𝑦 = 𝑎𝑡, where 𝑎 ∈ 𝑅, 𝑡 ∈ 𝑇 such that 𝑎

𝑛𝑡 ∈ 𝐿 for some 

𝑛 ∈ 𝑍+. But 𝐿 is an app-semi-prime submodule of 𝑇, then 𝑎𝑡 ∈ 𝐿 + 𝑠𝑜𝑐(𝑇), that is 𝑦 ∈ 𝐿 +

𝑠𝑜𝑐(𝑇) so 𝐸𝑇(𝐿) ⊆ 𝐿 + 𝑠𝑜𝑐(𝑇). 

      (
         
⇐  ) Suppose that 𝑎𝑛𝑡 ∈ 𝐿, where 𝑎 ∈ 𝑅, 𝑡 ∈ 𝑇 and 𝑛 ∈ 𝑍+. Since 𝑎𝑛𝑡 ∈ 𝐿 then 

𝑎𝑡 ∈ 𝐸𝑇(𝐿) ⊆ 𝐿 + 𝑠𝑜𝑐(𝑇) by hypothesis. It follows that 𝑟𝑡 ∈ 𝐿 + 𝑠𝑜𝑐(𝑇). Hence 𝐿 is an app-

semi-prime submodule of 𝑇. 
 

Proposition (29) 

     Let 𝐿 be a proper submodule of an 𝑅-module 𝑇 with 𝑠𝑜𝑐(𝑇) ⊆ 𝐿. Then 𝐿 is an app-semi-

prime submodule of 𝑇 if and only if [𝐿:𝑅 𝑇] is a semi-prime ideal of 𝑅. 

Proof 

      (
        
⇒ ) Let 𝑎 ∈ √[𝐿:𝑅 𝑇], implies that 𝑎

𝑛 ∈ [𝐿:𝑅 𝑇] for some 𝑛 ∈ 𝑍
+, it follows that 

𝑎𝑛𝑇 ⊆ 𝐿, that is 𝑎𝑛𝑡 ∈ 𝐿 for all 𝑡 ∈ 𝑇. But 𝐿 is an app-semi-prime submodule of 𝑇, then 

𝑎𝑡 ∈ 𝐿 + 𝑠𝑜𝑐(𝑇) for all 𝑡 ∈ 𝑇. That is 𝑎𝑇 ⊆ 𝐿 + 𝑠𝑜𝑐(𝑇). Since 𝑠𝑜𝑐(𝑇) ⊆ 𝐿 we get 𝑎𝑇 ⊆ 𝐿 

since 𝐿 + 𝑠𝑜𝑐(𝑇) = 𝐿. That is 𝑎 ∈ [𝐿:𝑅 𝑇], hence √[𝐿:𝑅 𝑇] ⊆ [𝐿:𝑅 𝑇]. But [𝐿:𝑅 𝑇] ⊆

√[𝐿:𝑅 𝑇], it follows that  √[𝐿:𝑅 𝑇] = [𝐿:𝑅 𝑇], hence [𝐿:𝑅 𝑇] is a semi-prime ideal of 𝑅. 

      (
         
⇐  ) Suppose that 𝑎𝑛𝑡 ∈ 𝐿, where 𝑎 ∈ 𝑅, 𝑡 ∈ 𝑇 and 𝑛 ∈ 𝑍+, it follows that 𝑎𝑛 ∈ [𝐿:𝑅 𝑇], 

implies that 𝑎 ∈ √[𝐿:𝑅 𝑇]. But [𝐿:𝑅 𝑇] is a semi-prime ideal of 𝑅, implies that √[𝐿:𝑅 𝑇] =

[𝐿:𝑅 𝑇], hence 𝑎 ∈ [𝐿:𝑅 𝑇], it follows that 𝑎𝑇 ⊆ 𝐿 = 𝐿 + 𝑠𝑜𝑐(𝑇). Thus 𝑎𝑡 ∈ 𝐿 + 𝑠𝑜𝑐(𝑇) for 

all 𝑡 ∈ 𝑇. Thus 𝐿 is an app-semi-prime submodule of 𝑇. 
 

Proposition (30) 

      Let 𝑇 = 𝑇1 ⊕ 𝑇2 be an 𝑅-module, where 𝑇1 and 𝑇2 𝑅-modules, and 𝐿 = 𝐿1 ⊕ 𝐿2 be a 

submodule of 𝑇, where 𝐿1 is a submodule of 𝑇1 and 𝐿2 is a submodule of 𝑇2 with 𝐿 ⊆

𝑠𝑜𝑐(𝑇) = 𝑠𝑜𝑐(𝑇1) ⊕ 𝑠𝑜𝑐(𝑇2). If 𝐿 is an app-semi-prime submodule of 𝑇, then 𝐿1 is an app-

semi-prime submodule of 𝑇1 and 𝐿2 is an app-semi-prime submodule of 𝑇2. 

Proof 

     Let 𝑎𝑛𝑡 ∈ 𝐿1, where 𝑎 ∈ 𝑅, 𝑡 ∈ 𝑇1 and 𝑛 ∈ 𝑍
+, it follows that 𝑎𝑛(𝑡,0) ∈ 𝐿. But 𝐿 is an 

app-semi-prime submodule of 𝑇, then 𝑎(𝑡,0) ∈ 𝐿 + 𝑠𝑜𝑐(𝑇). But 𝐿 ⊆ 𝑠𝑜𝑐(𝑇), implies that 

𝐿 + 𝑠𝑜𝑐(𝑇) = 𝑠𝑜𝑐(𝑇),  thus 𝑎(𝑡,0) ∈ 𝑠𝑜𝑐(𝑇) = 𝑠𝑜𝑐(𝑇1) ⊕ 𝑠𝑜𝑐(𝑇2), it follows that 𝑎𝑡 ∈

𝑠𝑜𝑐(𝑇1) ⊆ 𝐿1 + 𝑠𝑜𝑐(𝑇1). Thus 𝐿1 is an app-semi-prime submodule of 𝑇1. 

Similarly we can prove 𝐿2 is an app-semi-prime submodule of 𝑇2. 
 

Proposition (31) 

      Let 𝑇 = 𝑇1 ⊕ 𝑇2 be an 𝑅-module, where each of  𝑇1 and 𝑇2 𝑅-module. Then the following 

statements are satisfy: 

1) 𝐿1 is an app-semi-prime submodule of 𝑇1 such that 𝐿1 ⊆ 𝑠𝑜𝑐(𝑇1) and 𝑇2 = 𝑠𝑜𝑐(𝑇2) if and 

only if 𝐿1 ⊕ 𝑇2 is an app-semi-prime submodule of 𝑇. 

2) 𝐿2 is an app-semi-prime submodule of 𝑇2 such that 𝐿2 ⊆ 𝑠𝑜𝑐(𝑇2) and 𝑇1 = 𝑠𝑜𝑐(𝑇1) if and 

only if 𝑇1 ⊕ 𝐿2 is an app-semi-prime submodule of 𝑇. 



  

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Proof 

1) (
        
⇒ ) Let 𝑎𝑛(𝑡1, 𝑡2) ∈ 𝐿1 ⊕ 𝑇2, where 𝑎 ∈ 𝑅, (𝑡1, 𝑡2) ∈ 𝑇 and 𝑛 ∈ 𝑍

+, then 𝑎𝑛𝑡1 ∈ 𝐿1. But 

𝐿1 is an app-semi-prime submodule of 𝑇1 and 𝐿1 ⊆ 𝑠𝑜𝑐(𝑇1), then 𝑎𝑡1 ∈ 𝐿1 + 𝑠𝑜𝑐(𝑇1) =

𝑠𝑜𝑐(𝑇1). Now, we have 𝑇2 = 𝑠𝑜𝑐(𝑇2) then 𝑎(𝑡1, 𝑡2) ∈ 𝑠𝑜𝑐(𝑇1) ⊕ 𝑠𝑜𝑐(𝑇2) = 𝑠𝑜𝑐(𝑇1  ⊕

𝑇2) ⊆ 𝐿1 ⊕ 𝑇2 + 𝑠𝑜𝑐(𝑇1  ⊕ 𝑇2). Thus 𝐿1 ⊕ 𝑇2 is an app-semi-prime submodule of 𝑇. 

  (
         
⇐  ) Suppose that 𝑎𝑛𝑡1 ∈ 𝐿1, where 𝑎 ∈ 𝑅, 𝑡1 ∈ 𝑇1 and 𝑛 ∈ 𝑍

+. Then for each 𝑡2 ∈ 𝑇2 

𝑎𝑛(𝑡1, 𝑡2) ∈ 𝐿1 ⊕ 𝑇2, but 𝐿1 ⊕ 𝑇2 is an app-semi-prime submodule of 𝑇, so 𝑎(𝑡1, 𝑡2) ∈ 𝐿1 ⊕

𝑇2 + 𝑠𝑜𝑐(𝑇1  ⊕ 𝑇2) = (𝐿1 ⊕ 𝑇2) + (𝑠𝑜𝑐(𝑇1) ⊕ 𝑠𝑜𝑐(𝑇2)). Since 𝐿1 ⊆ 𝑠𝑜𝑐(𝑇1) and 𝑇2 =

𝑠𝑜𝑐(𝑇2), it follows that 𝑎(𝑡1, 𝑡2) ∈ (𝐿1 ⊕ 𝑇2) + ((𝐿1 + 𝑠𝑜𝑐(𝑇1)) ⊕ 𝑇2), implies that 

𝑎(𝑡1, 𝑡2) ∈

(𝐿1 + 𝑠𝑜𝑐(𝑇1)) ⊕ 𝑇2 [because 𝐿1 ⊕ 𝑇2 ⊆ (𝐿1 + 𝑠𝑜𝑐(𝑇1)) ⊕ 𝑇2 implies that (𝐿1 ⊕ 𝑇2) +

((𝐿1 + 𝑠𝑜𝑐(𝑇1)) ⊕ 𝑇2) = (𝐿1 + 𝑠𝑜𝑐(𝑇1)) ⊕ 𝑇2]. Thus 𝑎𝑡1 ∈ 𝐿1 + 𝑠𝑜𝑐(𝑇1). Hence 𝐿1 is an 

app-semi-prime submodule of 𝑇1. 

2) Similarly we can prove (2). 
 

Proposition (32) 

      Let 𝑓:𝑇
          
→  𝑇′ be an 𝑅-epimorphism and 𝐿 is an app-semi-prime submodule of 𝑇′. Then 

𝑓−1(𝐿) is an app-semi-prime submodule of 𝑇. 

Proof 

       It is clear that 𝑓−1(𝐿) is a proper submodule of 𝑇. Now, let 𝑎𝑛𝑡 ∈ 𝑓−1(𝐿), where 𝑎 ∈ 𝑅, 

𝑡 ∈ 𝑇 and 𝑛 ∈ 𝑍+,  implies that  𝑎𝑛𝑓(𝑡) ∈ 𝐿. But L is an app-semi-prime submodule of 𝑇′, 

implies that 𝑎𝑓(𝑡) ∈ 𝐿 + 𝑠𝑜𝑐(𝑇′), it follows that  𝑎𝑡 ∈ 𝑓−1(𝐿) + 𝑓−1(𝑠𝑜𝑐(𝑇′)) ⊆ 𝑓−1(𝐿) +

𝑠𝑜𝑐(𝑇). Thus 𝑎𝑡 ∈ 𝑓−1(𝐿) + 𝑠𝑜𝑐(𝑇). Therefore 𝑓−1(𝐿) is an app-semi-prime submodule of  

𝑇. 
 

Proposition (33)  

      Let 𝑓:𝑇
          
→  𝑇′ be an 𝑅-epimorphism and 𝐾 be an app-semi-prime submodule of 𝑇 with 

𝐾𝑒𝑟 𝑓 ⊆ 𝐾. Then 𝑓(𝐾) is an app-quasi-prime submodule of  𝑇′. 

Proof  

      𝑓(𝐾) is a proper submodule of 𝑇′. If not,  𝑓(𝐾) = 𝑇′, that is 𝑡 ∈ 𝑇, then 𝑓(𝑡) ∈ 𝑇′ =

𝑓(𝐾), implies that 𝑓(𝑡) = 𝑓(𝑘) for some 𝑘 ∈ 𝐾, that is 𝑓(𝑡 − 𝑘) = 0, thus 𝑡 − 𝑘 ∈ 𝐾𝑒𝑟 𝑓 ⊆

𝐾, it follows that 𝑡 ∈ 𝐾, that is 𝑇 ⊆ 𝐾, but 𝐾 ⊆ 𝑇, so 𝑇 = 𝐾 contradiction. Now let  𝑎𝑛𝑡′ ∈

𝑓(𝐾), where 𝑎 ∈ 𝑅, 𝑡′ ∈ 𝑇′ and 𝑛 ∈ 𝑍+. But 𝑓 is an epimorphism, then 𝑓(𝑡) = 𝑡′ for some 

𝑡 ∈ 𝑇. That is 𝑎𝑛𝑓(𝑡) ∈ 𝑓(𝐾), implies that 𝑎𝑛𝑓(𝑡) = 𝑓(𝑘) for some 𝑘 ∈ 𝐾. That is 𝑓(𝑎𝑛𝑡 −

𝑘) = 0, it follows that 𝑎𝑛𝑡 − 𝑘 ∈ 𝐾𝑒𝑟 𝑓 ⊆ 𝐾, hence 𝑎𝑛𝑡 ∈ 𝐾. But 𝐾 is an app-semi-prime 

submodule of 𝑇, then 𝑎𝑡 ∈ 𝐾 + 𝑠𝑜𝑐(𝑇). Thus (𝑡) ∈ 𝑓(𝐾) + 𝑓(𝑠𝑜𝑐(𝑇)) ⊆ 𝑓(𝐾) +𝑠𝑜𝑐(𝑇′). 

That is 𝑎𝑡′ ∈ 𝑓(𝐾) + 𝑠𝑜𝑐(𝑇′). Hence 𝑓(𝐾) is an app-semi-prime submodule of 𝑇′. 

 
 

3. Conclusion 

      In this paper we define the concept of approximaitly semi-prime (for short app-semi-

prime) submodules, and we introduced several properties, characterizations of it. Also, we 

investigate the relationships of app-semi-prime submodules with prime submodules, semi-



  

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prime submodules, quasi-prime submodules and approximaitly prime submodules, we proved 

that app-semi-prime submodules are generalizations of above concepts, and we illistright the 

convers by examples. Also, we show by example that the resudule of an app-semi-prime 

submodule is not necessary app-semi-prime ideal of 𝑅, but we prove under certain conditions 

they are equivalents. A mange the main results we get are the following. 

1) Let 𝐿 be a proper submodule of an 𝑅-module 𝑇. Then 𝐿 + 𝑠𝑜𝑐(𝑇) is an app-semi-prime 

submodule of 𝑇 if and only if [𝐿 + 𝑠𝑜𝑐(𝑇):𝑇] is a semi-prime ideal of 𝑅 (hence an app-semi-

prime). 

2) Let 𝐿 be a proper submodule of an 𝑅-module 𝑇 with 𝑠𝑜𝑐(𝑇) ⊆ 𝐿. Then 𝐿 is an app-semi-

prime submodule of 𝑇 if and only if  [𝐿:𝑇 𝑎
𝑛] = [𝐿:𝑇 𝑎] for 𝑎 ∈ 𝑅 and some 𝑛 ∈ 𝑍

+. 

3) Let 𝐿 and 𝐸 are two app-semi-prime submodules of an 𝑅-module 𝑇 with 𝑠𝑜𝑐(𝑇) ⊆ 𝐿 or 

𝑠𝑜𝑐(𝑇) ⊆ 𝐸. Then 𝐿 ∩ 𝐸 is an app-semi-prime submodule of 𝑇. 

4) Let 𝐿 be a proper submodule of a multiplication 𝑅-module 𝑇. Then 𝐿 is an app-semi-prime 

submodule of 𝑇 if and only if 𝐾𝑛𝐹 ⊆ 𝐿 implies that 𝐾𝐹 ⊆ 𝐿 + 𝑠𝑜𝑐(𝑇), where 𝐾,𝐹 are 

submodules of  𝑇, 𝑛 ∈ 𝑍+. 

5) Let 𝐿 be a proper submodule of a multiplication 𝑅-module 𝑇. Then the following 

statements are equivalent: 

1)  𝐿 is an app-semi-prime submodule of  𝑇. 

2)  𝑡𝑛 ∈ 𝐿 implies that 𝑡 ∈ 𝐿 + 𝑠𝑜𝑐(𝑇) for every 𝑡 ∈ 𝑇. 

3)  √𝐿 ⊆ 𝐿 + 𝑠𝑜𝑐(𝑇). 

4) 𝐹1𝐹2 ……𝐹𝑗 ⊆ 𝐿, implies that 𝐹1 ∩ 𝐹2 ∩ ……∩ 𝐹𝑗 ⊆ 𝐿 + 𝑠𝑜𝑐(𝑇) for every submodules 

𝐹1,𝐹2,……,𝐹𝑗 of 𝑇 and 𝑗 ∈ 𝑍
+. 

6) Let 𝑇 be a faithful multiplication 𝑅-module and 𝐿 be a proper submodule of 𝑇. Then 𝐿 is an 

app-semi-prime submodule of 𝑇 if and only if [𝐿:𝑅 𝑇] is an app-semi-prime ideal of 𝑅. 

7) Let 𝑇 be a faithful finitely generated multiplication 𝑅-module and 𝐽 be an app-semi-prime 

ideal of 𝑅. Then 𝐽𝑇 is an app-semi-prime submodule of 𝑇. 

8) Let 𝑇 be a faithful finitely generated multiplication 𝑅-module and 𝐿 be a proper submodule 

of 𝑇. Then the following statements are equivalent. 

1) 𝐿 is an app-semi-prime submodule of 𝑇. 

2) [𝐿:𝑅 𝑇] is an app-semi-prime ideal of 𝑅. 

3) 𝐿 = 𝐼𝑇 for some app-semi-prime ideal 𝐼 of 𝑅. 

9) Let 𝑇 be non-singular finitely generated multiplication 𝑅-module and 𝐿 be a proper 

submodule of 𝑇. Then the following statements are equivalent. 

1) 𝐿 is an app-semi-prime submodule of 𝑇. 

2) [𝐿:𝑅 𝑇] is an app-semi-prime ideal of 𝑅.  

3) 𝐿 = 𝐼𝑇 for some app-semi-prime ideal 𝐼 of 𝑅 with 𝑎𝑛𝑛(𝑇) ⊆ 𝐼. 

10) Let 𝑇 = 𝑇1 ⊕ 𝑇2 be an 𝑅-module, where 𝑇1 and 𝑇2 𝑅-modules, and 𝐿 = 𝐿1 ⊕ 𝐿2 be a 

submodule of 𝑇, where 𝐿1 is a submodule of 𝑇1 and 𝐿2 is a submodule of 𝑇2 with 𝐿 ⊆

𝑠𝑜𝑐(𝑇) = 𝑠𝑜𝑐(𝑇1) ⊕ 𝑠𝑜𝑐(𝑇2). If 𝐿 is an app-semi-prime submodule of 𝑇, then 𝐿1 is an app-

semi-prime submodule of 𝑇1 and 𝐿2 is an app-semi-prime submodule of 𝑇2. 

11) Let 𝑇 = 𝑇1 ⊕ 𝑇2 be an 𝑅-module, where each of  𝑇1 and 𝑇2 𝑅-module. Then the 

following statements are satisfy: 

1) 𝐿1 is an app-semi-prime submodule of 𝑇1 such that 𝐿1 ⊆ 𝑠𝑜𝑐(𝑇1) and 𝑇2 = 𝑠𝑜𝑐(𝑇2) if and 

only if 𝐿1 ⊕ 𝑇2 is an app-semi-prime submodule of 𝑇. 



  

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Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (3) 2019 

 

2) 𝐿2 is an app-semi-prime submodule of 𝑇2 such that 𝐿2 ⊆ 𝑠𝑜𝑐(𝑇2) and 𝑇1 = 𝑠𝑜𝑐(𝑇1) if and 

only if 𝑇1 ⊕ 𝐿2 is an app-semi-prime submodule of 𝑇. 

12) Let 𝑓:𝑇
          
→  𝑇′ be an 𝑅-epimorphism and 𝐿 is an app-semi-prime submodule of 𝑇′. Then 

𝑓−1(𝐿) is an app-semi-prime submodule of 𝑇. 

13) Let 𝑓:𝑇
          
→  𝑇′ be an 𝑅-epimorphism and 𝐾 be an app-semi-prime submodule of 𝑇 with 

𝐾𝑒𝑟 𝑓 ⊆ 𝐾. Then 𝑓(𝐾) is an app-quasi-prime submodule of  𝑇′. 
 

References 

1. Lu, C.P. Prime Submodules of Modules.  Comm. Math. University Sancti Pauli.1984, 33, 

61-69. 

2. Haibat, K.M.; Ali, S.h.A. Approximaitly Prime submodules and Some Related concepts. 

Ibn AL-Haitham Journal for Pure and Applied Science.2019, 32, 2, 103-113. 

3. Haibat,   K.M.;  Saif, A.H.WE-Prime submodules and WE-semi-prime submodules. Ibn-

Al-Haitham Journal for Pure and Applied Science.2018, 31, 3, 109-117. 

4. Haibat, K.M.; Wissam, A.H. WN-2-Absorbing and WES- 2-Absorbing submodules. Ibn 

AL-Haitham Journal for Pure and Applied Science.2018, 31, 3, 118-125. 

5. Haibat, K.M.;  Omer,  A.A. Pseudo Quasi2-Absorbing submodules and Some Related 

concepts. Ibn AL-Haitham Journal for Pure and Applied Science.2019, 32, 2. 

6. Dauns, J. Prime Modules. Journal reine Angew, Math.1978, 2, 156-181. 

7. Athab, E.A. Prime and Semi prime Submodules, Ph.D. thesis; College of Science, 

University of Baghdad,1996. 

8. Abdul – Razak, H.M. Quasi-Prime Modules and Quasi-Prime Submodules. M.Sc. Thesis, 

University of Baghdad,1999. 

9. Farzalipour, F. On Almost Semi-prime Submodules. Hindawi Algebra.2014, 752858, 1-6. 

10. Nuhad, S.A.; Al-Hakeem, M.B. Nearly Semi-prime submodules. Iraqi Journal of 

Science.2015, 56, 4B, 3210-3214. 

11. Kasch, F. Modules and Rings. London Math. Soc. Monographs (17), New York, 1982. 

12. Gooderal, K.R. Ring Theory, Nonsingular Ring and Modules; Marcel. Dekker, New 

York,1976. 

13. Anderson, F.W.; Fuller, K.R. Rings and Categories of Modules; Springer-Verlag, New 

York,1992. 

14. Abd El-Bast, Z.; Smith, P.F. Multiplication Modules.  Comm. Algebra.1988, 16, 4, 755-

779. 

15. Soheilnia, F.; Yousefian, A. 2-Absorbing and weakly 2-Absorbing Submodules. Thai 

Journal of Math.2011, 3, 577-584.  

16. Smith, P.F. Some Remarks on Multiplication Module. Arch. Math.1988, 50, 223-225. 

17. Hamid, A.T.; Rezva, V. Semi-Radical of Submodules in Modules. International Journal 

of Engeneering Science.2008, 19, 1, 21-27.