56 

 

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

 

 

  

 

 

 

Weakly Approximaitly Quasi-Prime Submodules And Related Concepts 

Haibat K. Mohammadali                                                   Shahad  J. Mahmood 

dr,mohammadali 2013@gmail.com                                    hasanenjassm@gmail.com 

Department Of Mathematics ,College Of Computer Sciences And Mathematics ,University Of Tikrit, 

Tikrit, Iraq 

 

 

 

Abstract  

           Let R be  commutative Ring , and let T be  unitary left R − module .In this paper 

,WAPP-quasi prime submodules are introduced as  new generalization of Weakly quasi prime 

submodules , where  proper submodule C of an R-module T is called WAPP –quasi prime 

submodule of T, if whenever 0≠rstϵC, for r, s ϵR , t ϵT, implies that either  r tϵ C +soc(T)  or  

s tϵC +soc(T) .Many examples of characterizations and basic properties are given . 

Furthermore several characterizations of WAPP-quasi prime submodules in the class of 

multiplication modules are established. 

 

Keywords: Weakly quasi prime submodules ,WAPP-quasi prime submodules , Socle of 

modules , Z-Regular modules , Projective modules . 

1. Introduction  

     Throughout this paper , all rings are commutative with identity , and all modules are left 

unitary R-modules . Weakly quasi prim submodules was first introduced  and studied in 2013 

by [1] as a generalization of a weakly prime submodule , where  proper submodule C of  R-

module T was called weakly prime submodule of C , if whenever  0≠atϵC  ,for a ϵR , t ϵT, 

implies that either tϵC or aT C [2] , and a proper submodule C of  R-module T is called  

weakly quasi prime submodule of T , if whenever 0≠abtϵC , for a ,b ϵR  t ϵT, implies  that 

either at ϵC  or bt ϵC . Recently many generalization of weakly quasi prime submodules were 

introduced see [3, 4, 5] . In this research we introduced another generalization of weakly quasi 

prime submodule , where  proper submodule C of  R-module T is called WAPP-quasi prime 

submodule of T , if whenever  0≠abtϵC for a, b ϵR , tϵT implies that either atϵC+soc(T) or bt 

ϵC +soc(T). Soc(T)  is the socle of a module T, defined by the intersection of all essential 

submodule of T [6] , where a nonzero submodule A of an R-module T is called essential if 

A∩ B ≠ (0)for   each nonzero submodule B of T [6] . Recall that R-module T is 

Ibn Al Haitham Journal for Pure and Applied Science 

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

 

Doi: 10.30526/34.2.2613 

 

Article history: Received, 21,April ,2020, Accepted 21,June,2020, Published in April 2021 

 

file:///C:/Users/المجلة/Desktop/العدد%20الثاني%20هدر/شهد.docx
mailto:2013@gmail.com
mailto:hasanenjassm@gmail.com


  

57 

 

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

 

multiplication if every submodule C of T is of the form IT for some ideal I of R , in particular 

C=[C:R T] T [7]. Let A and B be a submodule of multiplication module T with A=IM and 

B=JT for some ideals I ,J of R , then AB=IJT=IB . In particular AT=ITT=IT=A. Also for any 

t ϵT , At=A< t >= It [8]. Recall   that an R-module T is faithful , if ann(T)= (0) [7] .A R-

module T is a projective if for any epimorphism f  from  R-module X into X ’ and for any 

homomorphism g from Tin to X’ there exists a homomorphism h from T in to X such that f o 

h=g [7].Recall that an R-module T is a Z-regular , if for each tϵT there exists fϵT*=Hom(T,R) 

such that t=f(t)t [10] 

2.Basic Properties of WAPP-Quasi Prime Submodule 

     In this section, we introduced the definition of WAPP-quasi prime submodules and 

established some of its basic properties , characterization and examples. 

Definition(1) 

     A proper submodule C of an R − module T is called Weakly approximaitly quasi prime 

submodule of T (for short WAPP-quasi prime submodule) , if whenever 0≠abt ϵC , for a, bϵR, 

tϵT,  implies that either atϵ C+Soc(T)  or btϵC+Soc(T). 

And an ideal J of  ring R is called WAPP-quasi prime ideal of R if J is  WAPP- quasi prime 

submodule of R-module T. 

Examples and Remarks(2) 

1. The submodule C=< 12̅̅̅̅ > of the Z-module Z24 is a WAPP-quasi prime submodule of 

Z24, , since Soc(Z24)=< 4̅ > , and for 0≠abtϵ< 12̅̅̅̅ > ={0̅ , 12̅̅̅̅ } for a,b ϵZ ,tϵZ24, 

implies that either atϵ< 12̅̅̅̅ >+Soc(Z24) or bt ϵ< 12̅̅̅̅ >+Soc(Z24) . That is either atϵ<

12̅̅̅̅ >+Soc(Z24)=< 4̅ > or btϵ< 12 >+Soc(Z24)=< 4̅ > thus 0≠2.3. 2̅ϵ< 12̅̅̅̅ > for 

2,3,ϵZ, 2̅ϵZ24 implies  that 2. 2̅=4̅ϵ  < 12̅̅̅̅ >+< 4̅ >=< 4̅ >={0̅, 4̅, 8̅, 12̅̅̅̅ , 16̅̅̅̅ , 20̅̅̅̅ } . 

2. The submodule 12Z of the Z-module Z is not WAPP-quasi prime submodule , since 

Soc(Z)=(0) and whenever 0≠3.4.1ϵ 12Z , for 3,4,1ϵZ , implies that 3.112Z+Soc(Z) 

and 4.112Z+Soc(Z) 

3.  It is clear that every weakly quasi prime submodule of an R-module T is WAPP-quasi 

prime but not conversely .  

The following example explains that : 

Consider the Z-module Z24, and the  submodule C=< 6̅ > ={0̅, 6̅, 12̅̅̅̅ , 18̅̅̅̅ } , C is not weakly 

quasi prime submodule of Z24 since 2.3.1̅ϵC =< 6̅ > , for 2,3 ϵZ , 1̅ϵZ24 , implies that 2. 

1̅=2̅≠< 6̅ > and 3. 1̅=3̅ < 6̅ > . Bat C is a WAPP-quasi prime submodule of Z24 , since 

Soc(Z24)=< 4̅ > , and whenever 0≠ abt ∈ C =    < 6̅ > = {0̅, 6̅, 12̅̅̅̅ , 18̅̅̅̅ } for a, b ∈ Z , t ∈Z24  

implies that either at∈ C + Soc(Z24)= < 6̅ > +< 4̅ >=< 2̅ > or bt∈ C + Soc(Z24)= < 6̅ >

+< 4̅ >=< 2̅ > .  

That is 0≠ 2.3. 1̅ ∈ C, for 2,3 ∈ Z , 1̅ ∈Z24 , implies that 2.1̅ ∈ C + Soc(Z24)=< 2̅ >.  

4. It is clear that ever weakly prime submodule of an R-module T is a WAAP-quasi 

prime but not conversely .  

The following example explains that :  



  

58 

 

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

 

Consider the Z-module Z24 and the submodule C=< 12̅̅̅̅ >={0̅, 12̅̅̅̅ }. From (1), C is WAPP-

quasi prime submodule of Z24. But C is not weakly prime submodule of Z24 . Since  if 0≠

3. 4̅ ∈ C , for 3ϵZ , 4̅ϵZ24 , but 4̅ C and 3[C:Z24]=6Z 

5. The residual of WAPP-quasi prime submodule C of an R-module T needs not to be 

WAPP-quasi prime ideal of R .  

The following example explains that : 

We have seen in(1) that the submodule C=< 12̅̅̅̅ > of the Z − module Z24 is a  WAPP-quasi 

prime but [C:Z Z24]=[< 12̅̅̅̅ >:Z Z24]=12Z is not WAPP-quasi prime ideal by (2).  

6. The submodules PZ of a Z-module Z is a WAPP-quasi prime if and only if P is prime 

number 

7. The intersection of two WAPP-quasi prime submodule of  R-module, T need, not to 

be  WAPP-quasi prime submodule of T for example:                                                                                          

The submodule 2Z and 5Z of the Z-module Z are WAPP-quasi prime submodule by (6). 

But 2Z∩5Z=10Z is not WAPP-quasi prime submodule of theZ − module Z , since 

0≠2.5.1ϵ10Z, for 2,5,1ϵZ but 2.1=210Z+Soc(Z) and 5.1=510Z+Soc(Z) 

The following proposition are characterizations of WAPP-quasi prime submodules . 

Proposition(3) 

Let 𝑇 be an R − modul and C be proper submodul of T , then C is  WAPP −

      quasi prime sub modul of T if and only if , whenever 0≠rsBC , for r, sϵ R, 

      B is submodul of T , implies that either r BC +Soc(T) or s BC +Soc(T). 

Proof: 

() Assum that C is AWPP-quasi prime submodule  of T and 0≠rsBC . For r, s ϵR , B is 

a submodule of T , with r B C+ Soc(T) and s B C +Soc(T), that is there exists a nonzero 

elements b1 ,b2 ϵB such that rb1 C +Soc(T) and s b2  C +Soc(T).Now 0≠rsb1 ϵC , and C is  

WAPP-quasi prime submodule and r b1 C +Soc(T) , implies that s b1 ϵC +Soc(T). Also 

0≠rsb2 ϵC , and C is a WAPP-quasi prime submodule of T, and sb2 C +Soc(T) .,implies that 

rb2ϵC+Soc(T). Again since 0≠rs(b1+b2)ϵC and C is WAPP-quasi prime submodule of T , 

implies that either r(b1+b2)ϵC +Soc(T) or s(b1+b2)ϵC +Soc(T)  . If r(b1+b2)ϵC +Soc(T) ,that is 

rb1+rb2ϵC+Soc(T) ,and since rb2ϵC+Soc(T), it follows that rb1ϵC+Soc(T) which is 

contradiction . If s(b1+b2)ϵC +Soc(T) , that is sb1+sb2ϵC+Soc(T) and since sb1ϵC+Soc(T), it 

follows that sb2ϵC+Soc(T) which is contradiction. Hence r BC +Soc(T) or s B C +Soc(T). 

() 

 Let 0≠rstϵC , for r, s ϵR , t ϵT , it follows that 0≠rs<t>C , so by hypothesis either 

r<t>C+ Soc(T) or s<t>C+ Soc(T)   . That is either r tϵ C+ Soc(T) or s tϵ C+ Soc(T) .Hence 

C is WAPP − quasi prime submodul of T. 

 



  

59 

 

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

 

Proposition(4) 

 Let𝑇 be R − module and C be proper submodule of T . Then C is  WAPP −

quasi prime submodul of T if and only if whenever 0≠IJB C , for I,J are ideals of R and B 

is a submodule of T , implies that either IBC +Soc(T) or JBC +Soc(T). 

Proof: 

 ()Assume that 0≠IJBC . For I,J are ideal of R  , B is a submodule of T , with 

IBC+Soc(T) and J BC +Soc(T), so there exists a nonzero elements b1 ,b2 ϵ B and a nonzero 

elements r ϵI , s ϵJ such that rb1 C +Soc(T) and sb2 C +Soc(T).Now 0≠rsb1 ϵC , and C is a 

WAPP-quasi prime submodule and rb1 C+Soc(T) , implies that sb1ϵ C+Soc(T). Also 

0≠rsb2ϵC , and C is a WAPP-quasi prime submodule of T, and sb2 C+Soc(T) .,implies that 

rb2ϵC+Soc(T). Again 0≠rs(b1+b2)ϵC and C is WAPP-quasi prime submodule of T , implies 

that either r(b1+b2)ϵC +Soc(T) or s(b1+b2)ϵC +Soc(T)  . If r(b1+b2)ϵC +Soc(T) ,that is 

rb1+rb2ϵC+Soc(T) ,and rb2ϵC+Soc(T), implies  that rb1ϵC+Soc(T)  contradiction . If 

s(b1+b2)ϵC +Soc(T) , that is sb1+sb2ϵC+Soc(T) and  sb1ϵC+Soc(T), implies that sb2ϵC+Soc(T) 

which is contradiction. Hence IBC+Soc(T) or JBC+Soc(T). 

() 

 Suppose that  0≠rstϵ C , for r, s ϵR , tϵT ,  that is 0≠<r><s><t>C , so by our 

assumption either (r)(t) C+ Soc(T) or (r)(t)  C+ Soc(T)   . That is either rt∈ C +

Soc(T) or st∈ C + Soc(T)  .Hence C is WAPP-quasi prime submodule of T. 

As a direct consequence of the above propositions, we get the following corollaries. 

Corollary(5) 

Let 𝑇 be R − module and C be proper submodule of T . Then C is  WAPP −

quasi prime submodule of T iff whenever 0≠rIt C , for r ϵR, I is an ideals of R and t∈ T, 

implies that either r t ϵC +Soc(T) or I t C +Soc(T). 

Corollary(6) 

Let 𝑇 be R − module and C be proper submodule of T . Then C is  WAPP −

quasi prime submodule of T iff whenever 0≠IJt C , for J, , I is an ideals of R  and tϵ T , 

implies that either J tC +Soc(T) or ItC +Soc(T). 

Corollary(7) 

 Let 𝑇 be R − module and C be proper submodule of T . Then C is  WAPP −

quasi prime submodul of T if and only if, for each r ϵR and every ideal I of R  and every 

submodule B of T, with  0≠rIB C , implies that either rBC +Soc(T) or IBC +Soc(T). 

 

 

Proposition(8) 



  

60 

 

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

 

Let 𝑇 be R − module and C be proper submodule of T . Then C is  WAPP −

quasi prime submodul of T if and only if  for each  r,s ϵR ,[C:rs][0:T rs]∪[C+ Soc(T)::T 

r]∪[C+Soc(T):T s] . 

 

Proof: 

( )Let  tϵ[C:T  rs] , implies that rstϵC .If rst=0 , then tϵ[0:T  rs], and hence tϵ[0:T 

rs]∪[C+ Soc(T)::T r]∪[C+ Soc(T)::T s] .Suppose that 0≠rstϵC and since C is WAPP-quasi 

prime submodule of T , it follows that either rt ϵC +Soc(T) or st ϵC +Soc(T) , implies that 

either tϵ[C+Soc(T):T r] or tϵ[C+Soc(T):T s] .That is tϵ[0:T rs]∪[C+ Soc(T)::T r]∪[C+ Soc(T)::T 

s] .Hence, [C:T rs][0:T rs]∪[C+ Soc(T)::T r]∪[C+ Soc(T):T s] . 

()Assume that 0≠rstϵC , for  r,sϵR , tϵT ,  implies that tϵ,[C:T rs][ [0:T rs]∪ [C+ Soc(T):+:T 

r]∪[C+ Soc(T):T s] . But 0≠rst , then t [0:T rs] , hence tϵ [C+ Soc(T):T r]∪[C+ Soc(T):T s] , it 

follows that  rtϵC+Soc(T) or stϵC+Soc(T).Hence, C is  WAPP-quasi prime submodule of T. 

 

Proposition(9) 

 Let 𝑇 be R − modul and C be proper submodul of T . Then C is  WAPP −

quase priem submodul of T iff  for every  rϵR , and tϵT  with rtC +Soc(T)  ,[C:R rt][0:R 

rt]∪[C+Soc(T):R t] 

Proof: 

()Suppose that C is  WAPP-quase  ,and let sϵ[C:R rt], implies that rstϵC  .If rst=0 then 

sϵ[0:R r], hence sϵ[0:R rt]∪[C+Soc(T):R t]. If 0≠rstϵC and C is a WAPP-quasi prime 

submodule of T and  rtC +Soc(T) ,then stϵC+ Soc(T) that is sϵ[C+ Soc(T):R t] . Hence 

sϵ[0:R rt]∪[C+ Soc(T):R t] . Thus ,[C:R rt][0:R rt]∪[C+ Soc(T):R t]. 

As a direct consequence of proposition (9) and proposition (3),we get the following corollary: 

Corollary(10) 

 Let 𝑇 be R − modul and C be proper submodul of T . Then C is  WAPP −

quase prim submodul of T iff  for every  rϵR , and any submodule B of T   with rBC 

+Soc(T)  ,[C:R rB][0:R rB]∪[C+Soc(T):R B] 

As a direct consequence of proposition (9) and proposition (4)we get the following corollary. 

Corollary(11) 

 Let 𝑇 be R − module and C be proper submodule of T . Then C is  WAPP −

quasi prime submodule of T iff  for every  ideal I of R , and every submodule B of T   with 

IBC +Soc(T) , ,[C:R IB][0:R IB]∪[C+Soc(T):R B]. 

Proposition(12) 



  

61 

 

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

 

 Let 𝑇 be R − module and C be proper submodule of T . Then  for every s, rϵR , and 

tϵT   ,[C:R rst][0:R rst]∪[C+Soc(T):R r t] ]∪[C+Soc(T):R s t]. 

Proof: 

    Suppose that eϵ[C:R rst] ,implies that rs(et)ϵC .If rs(et)=0, implies that eϵ[0:R rst] and hence 

eϵ[0:R rst]∪[C+Soc(T):R r t] ]∪[C+Soc(T):R s t].If rs(et)≠0 ,and C is a WAPP-quasi prime 

submodule of T , then either r(et)ϵC+Soc(T) or s(et)ϵC+Soc(T). That is either 

eϵ[C+Soc(T):Rrt] or eϵ[C+Soc(T):Rst] thus eϵ[0:R rst]∪[C + Soc(T):R r t] ]∪[ C + Soc(T):R s 

t].Therefore ,[C:R rst][0:R rst]∪[ C + Soc(T):R r t] ]∪[ C + Soc(T):R s t]. 

The following are characterizations in the multiplication module . 

Proposition(13) 

     Let T be  multiplcation  R_module  and C be proper submodule  of T . Then C is a WAPP −

quasi prime submodule of T iff  0≠K1K2t C , for some  submodules K1 ,K2 of T , and tϵT 

implies that either K1tC +Soc(T) or K2tC +Soc(T). 

Proof: 

() Suppos that C is  WAPP −  quasi prime submodul of T , and 0≠K1K2t C for some  

submodules K1 ,K2 of T , and tϵT . Since T is a multiplication , then K1=IT and K2=JT 

for some ideals I,J of R .Thus 0≠K1K2t=IJt C. Since C is a WAPP-quasi  prime submodule 

of T then by corollary (6) either ItC + Soc(T) or Jt C + Soc(T). Hence either K1t C +

Soc(T)  or K2t C + Soc(T). 

() Assume that 0≠IJtC , for some ideals  I,J of R ,tϵT .That is 0≠K1K2t C for  K1=IT and 

K2=JT . It follows that either  K1tC+Soc(T) or K2tC+Soc(T); that is  It C + Soc(T)  or 

Jt C + Soc(T).Hence C is a WAPP-quasi prime  submodule of T by corollary(6). 

Proposition(14) 

Let T be  multiplcation  R_module  and C be proper submodule  of T . Then C is  WAPP −

quasi prime submodule of T iff 0≠K1K2H C , for some  submodules K1 ,K2  and H of T , 

implies that either K1HC +Soc(T) or K2HC +Soc(T). 

Proof: 

() Assume that 0≠K1K2H C for some  submodules K1 ,K2  and H of T . Since T is a 

multiplication , then K1=IT ,K2=JT for some ideals I,J  of R hence  0≠K1K2H=IJH C. But C 

is  WAPP-quasi prime submodule of T then by proposition (4) either IH C + Soc(T). or 

JH C + Soc(T).. Hence either K1H C + Soc(T). or K2H C + Soc(T).. 

() Let 0≠IJHC , where  I, J  are ideals of R , and H is a submodule of T .Since T is 

multiplication , then 0≠IJH=K1K2HC , hence by assumption either K1H C + Soc(T) or 

K2H C + Soc(T).That is either IH C + Soc(T) or JH C + Soc(T). Thus by proposition 

(4)C is  WAPP-quasi prime submodul of T . 



  

62 

 

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

 

It is well − knwon that if T is  Z − regular R − module , then Soc(T)=Soc(R)T [11;prop.(3-

25)] . 

Proposition(15)  

     Let T be Z_regular  multiplcation  R_module  and C be proper submodule  of T . Then C is WAPP −

quasi prime submodul of T iff [C:R T] is  WAPP- quasi prime ideal of R.   

proof:  

()Suppose that C is  WAPP − quasi prime submodule of T and let 0≠abI[C:R T] ,for 

a,bϵR , I is an ideal of R .it follows that 0≠ab(IT)C . Since C is  WAPP- quasi prime 

submodule of T , then by proposition(3) either aITC+Soc(T) or bITC+Soc(T).But T is a Z 

–regular module , then Soc(T)=Soc(R)T ,and since T is multiplication , then C=[C:R T]T . 

Hence  either aIT[C:RT]T+Soc(R)T  or bIT[C:RT]T+Soc(R)T. Thus  either 

aI[C:RT]+Soc(R)  or bI[C:RT]+Soc(R) . Hence by proposition [C:RT] is a WAPP-

quase priem ideal of R.   

()Suppose that [C:RT] is a WAPP-quasi  prime ideal of R , and 0≠rsB C , for r,sϵR , and B 

is a submodule of T . Since T is a multiplication , then B=IT ,for some ideal I of R ,that is 

0≠rsI TC, it follows that0≠rsI[C:R T]  . For [C:RT] is a WAPP-quasi prime ideal , then by 

proposition(3) either  rI[C:RT]+Soc(R)  or sI[C:RT]+Soc(R) , it follows that either 

rIT[C:RT]T+Soc(R)T  or sIT[C:RT]T+Soc(R)T. But T is a Z-regular Soc(T)=Soc(R)T and 

since T is a multiplication , then [C:RT]T=C .Thus  either rBC+Soc(T) or 

sBC+Soc(T).Hence by proposition (3) C is a WAPP-quasi prime submodule of T.  

It is well-known that if  an R-module T is  projective , then Soc(T)=Soc(R)T [11;prop.(3-24)]  

 Proposition(16) 

 Let T be a projective  multiplication  R-module and C be a proper submodule of T . 

Then C is  WAPP-quasi prime submodule of T if and only if [C:R T] is a WAPP- quasi prime 

ideal of R. 

Proof: 

()Let 0≠rIJ[C:R T] ,for rϵR , I ,J are  ideal of R .then 0≠r I(JT)C . Since C is  WAPP- 

quasi prime submodule of T , then by corollary(7) either r(JT)C+Soc(T) or 

I(JT)C+Soc(T).Now  since T is a projective module , then Soc(T)=Soc(R)T ,and since T is 

multiplication , then C=[C:R T]T . Hence  either r(JT)[C:RT]T+Soc(R)T  or 

I(JT)[C:RT]T+Soc(R)T. It follows that , either rJ[C:RT]+Soc(R)  or IJ[C:RT]+Soc(R) . 

Hence by corollary(7)  [C:RT] is a WAPP-quasi prime ideal of R.  

()Let T 0≠rIB C , for r ϵR ,I is an ideal in R, and B is  submodule of. Since T is a 

multiplication , then B=JT ,for some ideal J of R .Thus  0≠rIJ TC, implies that 0≠rIJ[C:R 

T]  . But [c:RT] is a WAPP-quasi prime ideal , then by corollary(7) either  rJ[C:RT]+Soc(R)  

or IJ[C:RT]+Soc(R) , that is either rJT[C:RT]T+Soc(R)T  or IJT[C:RT]T+Soc(R)T. Since 

T is a projective then Soc(T)=Soc(R)T and since T is a multiplication , then [C:RT]T=C .Thus  



  

63 

 

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

 

either rBC+Soc(T) or IBC+Soc(T).Hence by corollary (7) C is a WAPP-quasi prime 

submodule of T.  

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

Lemma(17)[12, coro, of theo, (9)] 

     Let T be a finitely generated multiplication R-module and I ,J are ideals of R . Then ITJT 

if and only if IJ+annR (T). 

Proposition(18) 

Let T be a  finitely generated multiplcation Z_regular  R_module and I is  WAPP −

quasi prime ideal of R with annR (T) I . Then IT  is an WAPP-quasi prime submodule of 

T.    

Proof: 

     Let 0≠I1 I2 B IT , for I1,I2 are is  ideals of R,and B is  submodul of T. Since T is a 

multiplication then B=J T for some ideal J of R. That is Let 0≠I1 I2 (J T) IT, it follows by 

lemma (17) 0≠I1 I2 J I+annR(T). But annR(T)I ,implies that  I+annR(T)=I. That is 0≠I1 I2 J 

I. But I is a WAPP-quasi prime ideal of R , then by proposition (4) either0≠I1 J I+Soc(R) or 

0≠ I2 J I+Soc(R). It follows that either0≠I1 J T IT+Soc(R)T or0≠ I2 J T IT+Soc(R)T. But 

T is a Z-regular then soc(R)T=Soc(T). Hence either 0≠I1 B IT+Soc(T) or0≠ I2 B 

IT+Soc(T). Thus by proposition (4) IT is  WAPP-quasi  prime submodule of T . 

Proposition(19) 

 Let T be a  finitely generated multiplication projective  R-module and I is a WAPP-

quasi prime ideal of R with annR (T) I . Then IT  is  WAPP-quasi prime submodule of T. 

Proof: 

     Let 0≠rI1 B IT , for rϵR ,I1 is an ideal of R, and B is  submodule of T. Since T is  

multiplication then B=JT for some ideal J of R. That is Let 0≠rI1 (J T) IT, it follows by 

lemma (17) 0≠rI1 J I+annR(T). But annR(T)I ,implies that  I+ annR(T)=I. Hence 0≠rI1 J I, 

and since I is WAPP-quasi prime ideal of R , then by corollary (7) either0≠I1 J I+Soc(R) 

or0≠ r J I+Soc(R).That is either 0≠I1 J T IT+Soc(R)T or0≠ r J T IT+Soc(R)T. But T is a 

projective  then soc(R)T=Soc(T). Thus either 0≠I1 B IT+Soc(T) or0≠ r B IT+Soc(T). 

Hence by corollary (7) IT is  WAPP-quasi prime submodule of T . 

It is well-known that cyclic R-module is multiplication [13], and since cyclic R-module is a 

finitely generated, we get the following corollaries: 

Corollary(20) 

     Let T be a cyclic Z-regular  R-module and I is  WAPP-quasi prime ideal of R with annR 

(T) I . Then IT  is  an WAPP-quasi prime submodule of T. 

Corollary(21) 



  

64 

 

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

 

 Let T be a cyclic projective  R-module and I is an WAPP-quasi prime ideal of R with 

annR (T) I . Then IT  is an WAPP-quasi prim submodule of T. 

It is well-known that if a submodule C of an R-module T is essential in T, then 

Soc(C)=Soc(T) [6, P.29]. 

Proposition(22) 

     Let T be  R-module ,and A,B are submodules of T with A B and B is an essential in T. If 

A  is an WAPP-quasi prime submodule of T , then A is a WAPP-quasi prime submodule of B. 

Proof: 

     Let 0≠rstϵA ,for r,sϵR ,tϵB, that is tϵT . Since A is a WAPP-quasi prime submodule of T , 

then either rtϵA +Soc(T) or stϵ A +Soc(T). But B is essential in T , then soc(B)=Soc(T). That 

is either rtϵA+Soc(B) or stϵA+Soc(B).Hence A is an WAPP-quasi prime submodule of B. 

Corollary(23) 

     Let T be  R-module ,and A,B are submodules of T with A B and Soc(T) Soc(B). Then 

A is a WAPP-quasi prime submodule of B. 

It well-known that if A is a submodule of an R-module T , then Soc(A)=A∩Soc(T) [9,lema 

2.3.15] 

Proposition(24) 

 Let T be  R − module , and A, B are submodules of T with B not contain in A ,and 

Soc(T) B. If A  is a WAPP-quasi prime submodule of T , then A∩ B is a WAPP-quasi prime 

submodule of B. 

Proof: 

 It is clear that A∩ B is an proper submodule of B .Now ,let 0≠rstϵ A∩ B, for r,sϵR ,tϵB, 

implies that 0≠rstϵ A, since A is a WAPP-quasi prime submodule of T ,then either 

rtϵA+Soc(T) or stϵA+Soc(T) ,hence either rtϵ(A+Soc(T) )∩ Bor stϵ(A+Soc(T))∩ B. Since 

Soc(T)B, then by module law  either rtϵ(A∩ B)+ (B∩ Soc(T)) or stϵ(A∩ B)+(B∩ Soc(T)). 

That is either rtϵ(A∩ B)+  Soc(B) or stϵ(A∩ B)+Soc(B).Thus A∩ B is a WAPP-quasi prime 

submodule of B. 

It well-known that for each  submodule A of an R-module T , then Soc(A)=A, then 

ASoc(T)[9,theo.(9.1.4)(c)]. 

 

Proposition(25) 

 Let T be an R − module , and A, B are submodules of T with B not contain in A, 

with Soc(A)=A and soc(B)=B. Then A∩ B is a WAPP-quasi prime sub module of T. 

Proof: 



  

65 

 

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

 

 Let 0≠rsL A ∩ B, for r,sϵR ,L is  submodule of T, then  0≠rs L  A ,and 0≠rs L B. 

But both  A ,B are  WAPP-quasi prime submodule of T, then either rLA+Soc(T) or 

sLA+Soc(T), and  rL B+Soc(T) or sL B+Soc(T). But Soc(A)=A and soc(B)=B , then 

ASoc(T) and BSoc(T), hence A+Soc(T)=Soc(T) and B+Soc(T)=Soc(T) ,A∩ B Soc(T) , 

implies that A∩ B + Soc(T)=Soc(T) ,so either rLSoc(T)= A∩ B + Soc(T) or sL Soc(T)= 

A∩ B + Soc(T). Hence A∩ B is  WAPP-quasi prime submodule of T 

Proposition(26) 

Let f:T→T′ be an R-epimorphism , and C be an  WAPP-quasi prime submodule of T with 

kerfC . Then f(C) is  WAPP-quasi prime submodule of T′. 

Proof: 

 Let f:T→T′ be an R-epimorphism , and C be an  WAPP-quasi prime submodule of T 

with kerfC ,let 0≠rst’ϵf(C) , for r,sϵR ,tϵT′ .Since f is onto , then f(t)= t , for some tϵT , it 

follows that 0≠rsf(t)ϵf(C), 0≠f(rst)ϵf(C) , so there exists a nonzero xϵC  such that, 

0≠f(rst)=f(x). That is f(rst-x) =0, implies that rst-xϵ kerfC, implies  that 0≠rstϵC. But C is a 

WAPP-quasi prime submodule of T , then either rtϵC+Soc(T) or   s tϵ C +Soc(T). That is 

either r f(t) ϵf(C)+f(Soc(T))  f(C)+Soc(T′) or sf(t)ϵf(C) +f(Soc (T))   f(C) +Soc(T′)  . Thus 

either r𝑡′ ϵ  f(C)+Soc(T′)  or s𝑡′ϵ  f(C) +Soc(T′)  . Hence f(C) is an  WAPP-quasi  prime 

submodule of T′ . 

 Proposition(27) 

Let f:T→T′ be an R-epimorphism , and C be  WAPP-quasi prime submodule of T′ . Then  

f −1(C) is an WAPP-quasi  prime submodule of T . 

Prove: 

 It is clearly that f −1(C)  is  proper submodule of T.  Let 0≠rstϵ f −1(C)  , for r,s ϵR ,tϵT  

, it follows that  then  0≠rsf(t)ϵC, but C is a WAPP-quasi  prime submodule of T , then either 

r f(t) ϵC +Soc(T) or sf(t)ϵ C +Soc(T). Thus either r tϵ f -1(C)+f -1 (Soc(T′)) f -1(C)+Soc(T) or  

s tϵ  f -1(C) + f −1 (Soc (T′))  f −1(C) +Soc(T) .Hence      f −1(C)  is  WAPP-quasi  prime 

submodule of T . 

 Proposition(28) 

 Let T be a Z-regular finitely generated multiplication R − module , and C be a proper 

submodule of T . Then the following statements are equivalent : 

1. C is  WAPP-quasi  prime submodule of T . 

2. [C:RT] is  WAPP-quasi  prime ideal of R . 

3. C=IT for some WAPP-quasi prime ideal I of R with annR(T)≤I . 

 

Poof: 

(1)  (2) Follows by proposition [15] 



  

66 

 

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

 

(2)  (3) Follows directly . 

(3)  (2) Suppose that C=IT for some a some WAPP-quasi prime ideal of R. Since T is  

multiplication , then C=[C:RT]T=IT and since M is finitely generated multiplication , then 

.[C:RT]= I+annR(T). But annR(T)I it follows that I+annR(T)=I. Thus [C:RT]=I is a WAPP-

quasi prime ideal of R. Hence [C:RT] is  WAPP-quasi prime ideal of R. 

The following corollary is a direct consequence of proposition (28) 

Corollary(29) 

 Let T be a cyclic  Z-regular  R-module , and C be  proper submodule of T . Then the 

following statements are equipollent : 

1. C is  WAPP-quasi  prime submodule of T . 

2. [C:RT] is  WAPP-quasi  prime ideal of R . 

3. C=IT for some WAPP-quasi  prime ideal I of R with annR(T)I . 

Proposition(30) 

 Let T be a finitely generated multiplication projective  R-module , and C be a proper 

submodule of T . Then the following statements are equipollent : 

1. C is a WAPP-quasi  prime submodule of T . 

2. [C:RT] is  WAPP-quasi  prime ideal of R . 

3. C=IT for some WAPP-quasi prime ideal  I of  R with  annR(T)I . 

Proof: 

(1)  (2) Follows by proposition (16) 

(2)  (3) Follows directly. 

(3)  (2) Follows as in proposition(28). 

As a direct consequence of proposition (30), we get the following corollary : 

Corollary(31) 

Let T be  cyclic  projctive  R − module , and C be  proper submodule of T , and C be a 

proper submodule of T . Then the following statements are equipollent : 

1. C is  WAPP-quasi  prime submodule of T . 

2. [C:RT] is  WAPP-quasi prime ideal of R . 

3. C=IT for some WAPP-quasi  prime ideal  I of  R with annR(T)I . 

It is well-known that if T is faithful multiplicationR − module , then Soc(T)=Soc(R)T 

[7,CORO.(2.14)(1)]. 

Proposition(32) 

 Let T be a faithful  multiplication  R − module and C be a proper submodule of T . 

Then C is a WAPP-quasi prime submodule of T iff  [C:R T] is a WAPP- quasi  prime ideal of 

R. 



  

67 

 

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

 

Proof: 

()Let 0≠IJk[C:R T] ,where I , J and k are  ideals of R .then 0≠ IJ(kT)C . Since C is  

WAPP- quasi  prime submodule of T , then by proposition(4) either J(kT)C+Soc(T) or 

J(kT)C+Soc(T).But  T is a faithful  multiplication ,it follows that C=[C:R T]T and  

Soc(T)=Soc(R)T . Thus  either I(KT)[C:RT]T +Soc(R)T  or J(KT)[C:RT]T +Soc(R)T. 

Hence either I K[C:RT]+Soc(R)  or JK[C:RT]+Soc(R) . Thus by proposition(4)  [C:RT] is  

WAPP-quasi  prime ideal of R.  

 

()Let T 0≠abB C , for a,b ϵR , and B is  submodule of T. Since T is  multiplication , then 

B=JT ,for some ideal J of R . Thus  0≠abJTC, it follows that 0≠abJ[C:R T]  . But [C:RT] is  

WAPP-quasi  prime ideal of R , then by proposition(3) either  aJ[C:RT]+Soc(R)  or 

bJ[C:RT]+Soc(R) , it follows that  either aJT[C:RT]T+Soc(R)T  or bJT[C:RT]T+Soc(R)T. 

But T is a  faithful  multiplication  R-module then  either aBC+Soc(T) or bB 

C+Soc(T).Thus by proposition (3) C is a WAPP-quasi  prime submodule of T. 

 

The following corollary is a direct consequence of proposition(32) 

Corollary(33) 

     Let T be a faithful  cyclic  R − module and C be a proper submodule of T . Then C is  

WAPP-quasi  prime submodule of T if and only if [C:R T] is a WAPP- quasi prime ideal of R. 

 

3.Conclusion 

     In this proper, we introduced and studied the concept WAPP-quasi prime submodule , and 

we established several examples , characterizations and basic properties of this concept . 

WAPP-quasi prime submodule is  generalization of a Weakly quasi prime submodule so we 

give example for converse . 

 

Among C,  the main results of this paper  are the  following: 

1. Proper submoduel C of  R-module T is  WAPP-quasi prime submodule of T iff whenever 

(0)≠rsBC , for r,sϵR, B is a submodule of T ,implies that either rBC+Soc(T) or 

sBC+Soc(T) 

2. Proper submodule C of  R-module T is  WAPP-quasi prime submodule  of T iff whenever 

(0)≠IJBC , for I,J are ideals of R, and B is  submodule of T ,implies that either IBC +

Soc(T). or JBC + Soc(T). 

3.  Proper submodule C of  R-module T is  WAPP-quasi prime submodule  of T iff for all 

r,sϵR , [c:T  rs][0:T rs]∪[C:dT r]∪[C:T s] 



  

68 

 

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

 

4. Proper submoduel C of  R-module T is  WAPP-quasi prime submodule  of T iff for all rϵR 

,tϵT with rtC + Soc(T) ,[c:T rt][0:T rt]∪[ C + Soc(T):T t]. 

5. Proper submodule C of  multiplication R-module T is  WAPP-quasi prime submodule of T 

iff whenever (0)≠K1K2tC , for some submodules K1,k2 of  T  and tϵT, implies that either 

K1tC+Soc(T) or K2tC+Soc(T) 

6.  Proper submodule C of  Z-regular  multiplication R-module T is a WAPP-quasi prime 

submodule of T iff [C:R T] is  WAPP-quasi prime ideal of R. 

7. Proper submodule C of  projective  multiplication R-module T is WAPP-quasi prim 

submodule of T iff [C:R T] is  WAPP-quasi prime ideal of R. 

8. If T is a cyclic a Z-regular R-module and I  is  WAPP-quasi prime ideal of R with 

annR(T)I. Then  IT is  WAPP-quasi submodule of T.  

 

 

References 

1. Waad,  K . H. ;Weakly Quasi prime Modules and Weakly Quasi prime submodules ; M. Sc. 

Thesis , university of Tikrit 2013. 

2. Behboodi ,M.; Koohy H .;Weakly prime Modules; Vietnam J. Math. , 2004 ,32 ,2, 185-195. 

3. Haibat,  K. M. ; Khalaf H.A. ; Weakly Quasi -2-Absobing Submodules ; Tikrit J. of pure 

sci .2018,23 ,7,101-104 . 

4. Haibat,  K . M. ; Saif,  A.H. ;WE-primary Submodule and WE-quasi prime Submodules ; 

Tikrit J. of pure swwci , 2019 ,24 ,1 ,68-102 . 

5. Haibat, K. M. ; Omer, A.A. ;Pseudo Quasi 2-Absorbing Submodules and Some Related 

Concepts ; Ibn AL-Hatham J . for pure and Applied sci .,2019,32, 2, 114-122. 

6. Gooderal, K.R. ;Ring Theory, Non Singular Ring and Modules; Marcel .Dekker , New York 

, 1976. 

7.  El-Bast ,  Z. A. ;Smith P.F. ;Multiplication Modules; Comm . In Algebra , 1988, 16, 4, 

755-779. 

8. Darani,  A.Y. ;Sohelnai, F. ; 2-Absorbing and Weakly 2-Absorbing Submodules ; Tahi 

Journal Math.2011 ,9 , 577-584 . 

9. Kasch, F. ; Modules and Rings ; London Math . Soc. Monographs , New-York , Academic 

press , 1982 . 

10. Zelmanowitz, J. M .;Regular Modules ; Trans .Amer. Math .soc .(1973),163 , 341-355. 

11. Nuha ,H. H. ;The Radicals of Modules ;M .Sc. Thesis , university of Bagdad , 1996. 



  

69 

 

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

 

12. Smith, P.F. ;Some Remarks on Multiplication Modules; Arch. Math . 1988, 50 ,223-223.