Quasi-Semiprime Modules

Suppose that A is an abelain ring with identity and B is a unitary (left) A-module. In this paper, we introduce a type of module, namely quasi-semiprime. A-module, whenever √[𝑁: 𝐵] is a prime ideal for proper submodule N of B, then B is called quasi -semiprime module, which is a generalization of quasi-Prime A-module, whenever ann A N is a prime ideal for proper submodule N of B, then B is quasi-prime module. A comprehensive study of these modules is given, and we study the relationship between quasi-semiprime modules and quasi-prime. We put the condition coprime over cosemiprime ring for the two cocept quasi-prime modules and quasi-semiprime modules, which are equivalent. The concepts of prime modules and quasi-semiprime modules are equivalent. The condition of anti-hopfain makes quasi-prime is quasi-semiprime A-module. Whenever B is cyclic, coprime C-module, where C is the ring, each ideal is semiprime, which implies quasi-prime, quasi-simepime, and annCB are prime ideals. If F is an epimorphism from B 1 → B 2 , whenever B1 is a quasi-prime module, it implies B2 is a quasi-prime A-module, and the inverse image of quasi-semiprime is a quasi-prime A-module.


Introduction
Suppose that W is a left A-module, where A is a ring with unity.An A-module B is said to be prime whenever annAB=annAN for each non-zero submodule Nof B, where annAB={a ∈ A;bx=0 for each b ∈ B} [1,2].Hasan in [3] introduced the concept of qasi-prime A-modules, which is a generalization of prime A-modules, where an A-module W is called quasi-prime modulles if and only if for each non-zero submodule N of W, annA N is a prime ideal.Annin [9] calls an A-module W a coprime (dual notion of prime modules) if annAW=annAW/A for every proper submodule A of W. In this paper, we study a generalization of the quasi-prime module which we called the Quasi-semiprime A-module if √annB/N = √[N: B] is a prime ideal for each submodule N of B. This paper consists of two sections.In section one; we study the basic properties of a quasi-semiprime A-module.In section two, we study the relation between quasi-semiprime Amodules and prime A-modules.

Definition (2.1)
B is said to be a quasi-semiprime A-module if √[N: W] is a prime ideal for the proper submodule N of B.

Examples and Remarks (2.2)
1-It is clear that Zn is a quasi-semiprime A-module if and only if n is a prime number.2-If n can be written as a product of two prime numbers, then Zn is a quasi-semiprime A-module.Proof: Let n=p1p2; p1, p2 be two prime numbers, so N1= (p1),N2=(p2), then√[( 3-Zp∞ is not a quasi-semiprime module, since we know that every submodule of Zp∞ is of the form (1/p n +Z), where n is a non-negative integer, so √[

Proposition (2.3)
Every proper submodule N of quasi-semiprime module is a quasi-semiprime module.

Proof:
Suppose N is a proper submodule of quasi-semiprime A-module W. Let K be a proper submodule Recall that whenever B ≅ B/N for all proper submodule Nof modules B, then we said that anonsimple A-module B anti-hopfian module [4,5].

Proposition (2.4)
Suppose that B is an anti-hopfian quasi-prime A-module, then B is quasi-semiprime.

Proof:
Since B is an anti-hopfian module, then B ≅ B/N for N be a proper submodule of B, so there exists an isomorphism function f: B→ B/N; f(b)=b+N for each b ∈ B, so it is easy to check that annAB=annAB/N, then by [6], every anti-hopfian A-module is a coprime E-module, where E=End(W) and by [5] so by [3] implies annAW is a prime ideal, so either a n ∈ annAW or b n ∈ annAB.Thus, either a ∈ √[: ] or b∈ √[: ], which means B is a quasi-semiprime A-module.
The condition anti-hopfian we cannot drop for example: Z6 is quasi-semiprime A-module by (2.2), while it is not quasi-prime by [3], and Z6 is not anti-hopfain by [6].

Prorosition (2.5)
Suppose B is a coprime A-module of quasi-prime, then B is quasi-semiprime A-module.

Proof
Let B be a quasi-prime A-module, then by [3], annAB is a prime ideal, but W is a coprime Amodule, so annAB/N is a prime ideal for each non-zero submodule N of B, which means√[: ] is a prime ideal.Thus, W is a quasi-semiprime A-module.
Recall that an ideal K of the ring A is called nil radical and denoted by√, and is defined by: √ ={a∈ A; a n ∈ K , for some n Z+} [8].

Not (2.6)
Suppose C is a ring where every ideal is nil radical, which we call cosemiprime ring.

Theorem M (2.7)
Suppose that B is a coprime C-module.The following statements are equivalent: 1) B is a Quasi-Prime Module.

Theorem (2.8)
Let B be a cyclic coprime C-module, then the following statements are equivalent: 3-annCB is a prime ideal.

Proposition (2.9)
Suppose that B is an A-Module and J is an Ideal Of A which that is contained in annAB/N where N is a submodule of B. Then, B is a quasi-semiprime A-module ⟷ B is a quasi-semiprime A/J-Module.

Proof:
Since B1 is a quasi-semiprime A-module, so if a n b n B1⊆ N1 for each a,b∈ A, then either a n B1⊆ N or b n B1⊆ N. Thus, f(a n b n B1)⊆ f(N1) since f is a homomorphism implies f(a n ).Thus, √ 2 :  2 is a prime ideal, which means B2 is a quasi-semiprime module.

Corollary (2.11)
The inverse image of the quasi-semiprime module is a quasi -semiprime module.

Quasi-Semi-Prime A-Module and Prime Module
Now, we turn our attention to the relationship between quasi-semiprime modules and prime modules.

Proposition (3.1)
Suppose B is a coprime A-module, then every prime A-module is a quasi-semiprime A-module.

Proof
It follows directly by from [3] and Propositions (2.5).
The next example shows that the converse of Proposition (3.1) is not valid in general.
Let Z6 as a Z -module is quasi-semiprime module by Examples and Remarks (2.2), while it is not a prime module [1].