2-Quasi-Prime Modules

We introduce in this paper, the notion of a 2-quasì-prime module as a generalization of quasi-prime module, we know that a module E over a ring R is called quasi-prime module, if (0) is quasi-prime submodule. Now, we say that a module E over ring R is a 2-quasi-prime module if (0) is 2-quasi-prime submodule, a proper submodule K of E is 2-quasi-prime submodule if whenever a, b ∈ R , 𝑥 ∈ W and abx ∈ K , then either a 2 x ∈ K or b 2 x ∈ K . Many results about these kinds of modules are obtained and proved, also, we will give a characterization of these kinds of modules

Montana Abdul-Razak introduced in [5] Quasi-prime module, which is a generalization of prime modules.We say E is Quasi-prime R-module if (0) is quasi-prime submodule of E. where a proper submodule H of E is called quasi-prime submodule if whenever ,  ∈ , x∈E,  ∈E, implies either  ∈  or  ∈ .As a generalization for this notion, we will introduce 2-quasi-prìme R-module, if (0) is 2-quasi-prime submodule, where H is called 2-quasi-prime submodule if whenever ,  ∈ , x∈E,  ∈E, implies either  2  ∈  or  2  ∈ , [6].In this paper, many properties for this new kind of modules are proved such that every 2-prime module is 2-quasi-prime but the converse is not true.

Basic Properties of 2-quasi-prime module
We introduce in this section the concept of a 2-quasi-prime R-modules and we give a characterization of these kinds of modules.

Definition (2.1).
Let E be a module over a ring R, E is called a 2-quasi-prime R-module if (0) is 2-quasiprime submodule of E. In the upcoming proposition, we give a characterization for 2-quasi-prime module.
12. The Z-module  = ⨁  is 2-quasi-prime module Proof: Since E is prime module by [5] and by ( 5) E is 2-quasi-prime R-module.We need the following proposition to prove proposition (2.6).
Proof: Let K be any submodule of ( 1 ,  2 ).To show that    is 2-prime ideal of R.

Corollary (2.8).
The endomorphism of 2-quasi-prime R-module is also a 2-quasi-prime Rmodule.Proof: clear.Recall that an integral domain is a non-zero commutative ring with no non-zero divisors Proposition (2.9).If E is an R-module and  =   () such that E is a 2-quasi-prime Smodule and S is commutative then S is an integral domain.Proof: Let ,  ∈  such that . = 0. Thus (.)() = 0 for all  ∈ .But E is a 2-quasiprime S-module, so (0) is a 2-quasi-prime S-submodule of E. Thus either  2 () = 0 or  2 () = 0 for all  ∈ .therefore either  = 0 or  = 0 and this means that S is an integral domain.The following corollary are immediate consequence of the last theorem.

Corollary (2.10). Let
Moreover, if R is a 2-quasi-prime ring and E is an R-module then E is not necessarily 2-quasiprime module, for example: consider the Z-module  6 : notice that Z is a 2-quasi-prime ring since (0) is 2-quasi-prime ideal) but  6 is not 2-quasi-prime module.

Proposition (2.12).
If E is a faithful multiplication R-module, then E is a 2-quasi-prime Rmodule if and only if R is a 2-quasi-prime ring.Proof: suppose E is a 2-quasi-prime R-module.To prove that R is 2-quasi-prime ring.Let J be a non-zero ideal of R since E is multiplication R-module so  =  is a non-zero submodule of E. Hence    is a 2-prime ideal of R because E is a 2-quasi-prime module.And since E is a faithful R-module, so    =    thus    is a 2-prime ideal and R is a 2-quasi-prime ring.Now, for the converse, suppose R is a 2-quasi-prime ring, we want prove that E is 2-quasi prime module: let H be a non-zero submodule of E. Since E is a multiplication R-module, then  = , for some ideal A of R.But E is faithful so    =    =   , which is 2-prime ideal.Therefore by proposition (2.2) E is 2-quasi-prime R-module.
Recall that an R-module E is uniform if the intersection of any two non-zero submodules of E is not zero [9] Remark (2.13).We notice that not every 2-quasi-prime R-module is uniform for example the Z-module ⨁ is prime and hence 2-quasi-prime Z-module.But it is not uniform if  = ⨁(0) and  = (0)⨁ then  ∩  = (0).

Theorem (2.14).
Let E be an R-module and let I be an ideal of R, which is contained in   .Then E is a 2-quasi-prime R-module if and only if E is a 2-quasi-prime   ⁄ −module.

Remark (2.15
).If R is 2-quasi-prime R-module, then I is 2-prime ideal of R, but this not implies that I is 2-quasi-prime ideal of R, as the following example shows consider  8 as  8 -module.One can prove easily that  8 as 8 -module is 2-quasi-prime  8 −module.Let  = {0 ̅ , 4 ̅ }, I is a 2-prime ideal of  8 , but it is not a 2-quasi-prime ideal of Now, we will see the direct sum of 2-quasi-prime R-modules need not to be 2-quasi-prime.

Conclusion
1. Let E be an R-module, then E is a 2-quasi-prime R-module if and only if () is 2-prime ideal for every non zero  ∈ .
3. The endomorphism of 2-quasi-prime R-module is also a 2-quasi-prime R-module.4. If E is an R-module and  =   () such that E is 2-quasi-prime S-module and S is commutative, the S is an integral domain 5.If E is a faithful multiplication R-module, then E is a 2-quasi-prime R-module if and only if R is a 2-quasi-prime ring.6.Let E be an R-module and let I be an ideal of R, which is contained in   .Then E is a 2-quasi-prime R-module if and only if E is a 2-quasi-prime   ⁄ −module.
Fatima and Alaa introduced in [3] a 2-prime module as a generalization of a prime module as the following we say E is 2-prime R-module if (0) is 2-prime submodule of E, where every ISSN: