Z-Nearly Prime Submodules

Let R be a commutative ring with identity and N be a proper submoduleNof R-moduLe Mis called prime if whenever rx ∈ N; r ∈ R , x ∈ M , implies either x ∈ Nor r ∈ [N: M]. In this paper we say that N is Z-nearly prime submodule of R-moduLe M, if wheneverf ∈ M = Hom M, R , x ∈ M such that f x ∙ x ∈ N, then either x ∈ N + J(M) or f x ∈ [N + J M : M], J(M) is the Jacobson radical of M. We prove some result of this type of submodules.


Introduction
Throughout this paper is commutative ring with one. And is a unitary -module.A proper submodule of anmodule is called a prime submodule if for each ∈ , ∈ , such that ∈ , then either ∈ or ∈ [ : ], where : = { : ∈ , ⊆ }, [1]. There are several generalization of the notion of prime submodules as Zprime. The definition of Z-prime is came in [2], as following: we say that a proper submodule of an -module is called -prime if for each ∈ , ∈ * = , , such that . ∈ implies that either ∈ or ∈ [ : ] . In [3] we study nearly prime as a generalization of prime submodules and they define a nearly prime submodules as follows: a proper submodule of an -module is called nearly prime, if whenever ∈ , ∈ , ∈ implies either ∈ + ( ) or ∈ + : .
In this article, we study -nearly prime submodules as generalization of Z-prime submodules, we give basic properties and we illustrate the relation between Z-nearly prime submodule and -prime submodule. It is clear that each Z-prime submodule is Z-nearly prime submdule, but the converse is not true in general. However, we gives a condition under which the two concepts one equivalent. And we study the Z-nearly prime submodules in other modules such as F-regular, injective, divisible, injective hull modules.

Z-Nearly Prime Submodules
In this section wedefine the concepts of Z-nearly prime submodule andinvestigate some properties. As generalization of Z-prime submodles,

Definition (2.1):
A proper submodule of -moduLe is called -nearly prime submodule of if whenever ∈ * = ( , ), ∈ such that . ∈ , then either ∈ + ( ) or ∈ + : ; ( ) is the Jacobson radical of . Specially, an ideal of a ring is -nearly prime ideal of if and only if is a Z-nearly prime submodule of R-module .

Remarks and examples (2.2):
1) It clear that every -prime submodule of an R-module is -nearly prime submoduleof ,but the converse is not true in general.

Proposition (2.3):
If is -nearly prime submodule of an -module , is any proper submodule of such that ⊈ and ( ) ⊆ , then ∩ is -nearly prime submodule in

Corollary (2.4):
Let be a good ring and be an -module , be any proper submodule of such that ⊈ , = , then ∩ is -nearly prime in

Proof:
Since ⊈ , then ∩ is a proper submodule in . Since be a good ring and = , then ( ) ⊆ , by above proposition ∩ is -nearly prime in .

Proposition (2.5):
Let and be two -nearly prime submodules of and either ( ) ⊆ or ( ) ⊆ , then ∩ is -nearly prime of

Proposition(2.6):
Let be a submodule of an -module . If[ + ( ): ] is a maximal ideal of , then is a − nearly prime submodule of .

Z-nearly Prime Submodules
In this section we study the Z-nearly prime submodules in others modules such as -regular, Z-regular, injective, divisible, injective hull modules. First we need the following proposition:

Proposition(3.1):
If is a Z-nearly Prime submodule of an -module and ( ) = 0, then is a Z-prime submodule of .

Proof:
It is clear.
Recall that an Rmodule is said to be F-regular if each submodule of is pure, [5].

Corollary(3.2):
If is aZ-nearlyprime submodule of an -module and is a F-regular, then is a Z-prime submoduleof .

Corollary (3.3):
If N is a Z -nearly Prime submodule of an -module and / ( ) is anF-regular ring for every 0  ∈ , then is a Z-prime submodule of .

Proof:
Since / ( ) is a regular ring for every 0  ∈ , then is an -regular R-module by [7].Hence the result follows by Corollary (3.2).

Corollary(3.4):
If is a Z-Prime submodule of Z-regular R-module , then is a primesubmodule of .

Proof:
Since is a -regular -module, then M is a F-regular Rmodule by [7].Hence the result follows by Corollary (3.2).

Corollary(3.5):
If is a Z-nearly prime submodule of an R-module and ( ) = ( ) ∩ for each submodule of , then is a Z-prime submodule of , [4]. Then the following is a consequence of corollary(3.5).

Corollary(3.6):
If is a Z-nearly prime submodule of an R-module and is a good ring, then is a Z-prime submodule of.
By using this concept,we have the following:

Proposition(3.7):
Let is and is a divisible R-module such that ( ) ≠ . If is a Z-nearly prime submoduleof , then is a Z-prime submodule of .

Corollary(3.8):
Let be an injective define on an integral domain and ( ) ≠ . If is a Z-nearlyPrime submodule, then is a Z-prime submoduleof .
Recall that a submodule of an R-module is said to be essential, if has non-trivial intersection with every non-zero submodule of , [4] .
Recall that an R-module is called an injective hull or injective envelope of a module if it is an essential extension of and an injective module, [4].
By using this concept,we can give the following result:

Corollary(3.9)
Let be an injective hull on an integral domain and ( ) ≠ . If is a Z-nearlyPrime submodule, then is a Z-prime submoduleof .

Corollary(3.10):
Let has no proper essential extensions R-module and ( ) ≠ . If is a Z-nearly Prime submodule, then is a Z-prime submoduleof M.