Weakly Approximaitly Quasi-Prime Submodules And Related Concepts

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.

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 Rmodule 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).

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=< 12 ̅̅̅̅ > of the Z − module Z24 is a WAPP-quasi prime but [C:Z Z24]=[< 12 ̅̅̅̅ >: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).

Proposition(3)
Let 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).

Proposition(4)
Let 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).
As a direct consequence of the above propositions, we get the following corollaries.

Corollary(5)
Let 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 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 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). As a direct consequence of proposition (9) and proposition (3),we get the following corollary:

Corollary(10)
Let 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: As a direct consequence of proposition (9) and proposition (4)we get the following corollary. 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 .

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) (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 WAPPquasi 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)
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.

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.

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:
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 ′ .

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 .

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 .
(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 WAPPquasi 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 :