On Co-Primarily Packed Modules

Let R be a commutative ring with 1, and M is a (left) R-module. We introduce the concept of coprimarily packed submodules as a proper submodule N of an R-module M which is said to be Coprimarily Packed Submodule. If N N  α α∈Λ ⊆ where Nα is a primary submodule of M for each α ∈Λ , then i n i N N M  α 1 = + ≠ for some n α α α 1 2 , , , ∈Λ  . When there exists β ∈Λ such that N N M β + ≠ ; N is called Strongly Coprimarily Packed submodule. In this paper, we list some basic properties of this concept. We end this paper by explaining the relations between p-compactly packed and coprimarily packed submodules, and also the relations between strongly p-compactly packed and strongly coprimarily packed submodules.


Introduction
Coprimely packed rings were introduced by Erdo˘gdu for the first time in [1].Al-Ani gave an analogous concept in modules [2], that is, a proper submodule N of an R-module M which is called Coprimely Packed.If where N α is a prime submodule of M for each α ∈ Λ , then In this paper, we discuss the situation where the union of a family of primary submodules of M is considered.
In [2], the concept of compactly packed modules was introduced.We generalized this concept to the concept of p-compactly packed modules in [3], that is, a proper submodule N of an R-module M which is called P-Compactly Packed.If for each family { } N α α∈Λ of primary submodules of M with N N α α∈Λ ⊆


, there ex- for some β ∈ Λ , then N is called Strongly P-Compactly Packed.A module M is said to be P-Compactly Packed (Strongly P-Compactly Packed), if every proper submodule of M is p-compactly packed (strongly p-compactly packed).
In this paper, we introduce the definitions of coprimarily packed and strongly coprimarily packed module and discuss some of their properties.We end this paper by explaining the relations between p-compactly packed and coprimarily packed submodules, and also the relations between strongly p-compactly packed and strongly coprimarily packed submodules.

Coprimarily Packed and Strongly Coprimarily Packed Submodules
In this section we introduce the definition of coprimarily packed and strongly coprimarily packed module and discuss some of their properties.

Definition
Let N be a proper submodule of an R-module M. N is said to be Coprimarily Packed Submodule if whenever where N α is a primary submodule of M for each α ∈ Λ , then A module M is called Coprimarily Packed (Strongly Coprimarily Packed) module if every proper submodule of M is coprimarily packed (strongly coprimarily packed) submodule.It is clear that every strongly coprimarily packed submodule is a coprimarily packed submodule.
In the following proposition, we discuss the behavior of strongly coprimarily packed module under homomorphism.

Let
for every primary submodule N of M, then M is a strongly coprimarily packed module if and only if M ′ is a strongly coprimarily packed module.
Proof.Suppose that M is a strongly coprimarily packed module and let


where N ′ is a proper submodule of M ′ and W α ′ is a primary submodule of M ′ for each α ∈ Λ , so ( ) ( ) ( ) is a primary submodule of M for each α ∈ Λ there exists β ∈ Λ such that ( ) ( ) and ( ) ( ) which is a contradiction.So M ′ is a strongly coprimarily packed module.
Conversely, suppose M ′ is a strongly coprimarily packed module and let N W α α∈Λ ⊆


, where N is a proper submodule of M and W α is a primary submodule of M for each α ∈ Λ .Hence ( ) ( ) and since ker f W α ⊆ for each α ∈ Λ , ( ) , since f is an epimorphism, there exists y M ∈ such that ( ) f y x = and there exists n N ∈ and u W β ∈ such that y n u = + .Then ( ) ( ) ≠ , so M is a strongly coprimarily packed modul.
The following proposition gives a characterization of strongly coprimarily packed submodules in a multiplication or finitely generated module.

Proposition
Let M be a finitely generated or multiplication R-module.A proper submodule N is strongly coprimarily packed if and only if whenever where L α is a maximal submodule of M for each α ∈ Λ then there exists β ∈ Λ such that N L β ⊆ .Proof.Suppose N is a strongly coprimarily packed submodule and let N L α α∈Λ ⊆


where L α is a maximal submodule of M for each α ∈ Λ , hence L α is a primary submodule, so there exists β ∈ Λ such that where N α is a primary submodule for each α ∈ Λ .There exists a maximal submodule L α that contains N α for each α ∈ Λ , hence N N L . By hypothesis, there ex- , thus N is a strongly coprimarily packed submodule.
Recall that an R-module M is called Bezout Module if every finitely generated submodule of M is cyclic.
In the following proposition we will give a characterization for strongly coprimarily packed multiplication module.

Proposition
Let M be a multiplication R-module.If one of the following holds: 1) M is a cyclic module.
2) R is a Bezout ring.
3) M is a Bezout module.
Then M is strongly coprimarily packed module if and only if every primary submodule is strongly coprimarily packed.
Proof.Let N be a proper submodule of a module M such that N N α α∈Λ ⊆


where W α is a maximal submodule of M for each α ∈ Λ , then by proposition (2.3), it is enough to show that there exists β ∈ Λ such that , since N is a submodule of a multiplication module, there exists a primary submodule L that contains N. By hypothesis, L is strongly coprimarily packed submodule and , so it is primary submodule.Thus by hypothesis K is strongly coprimarily packed and since K W α α∈Λ ⊆  and by proposition (2.3) there exists β ∈ Λ such that N K W β ⊆ ⊆ .We end this Paper by looking at the relations between the strongly p-compactly packed modules and strongly coprimarily packed modules.
Recall that a proper submodule N of an R-module M is called P-Compactly Packed if for each family { } It is easy to show that every strongly p-compactly packed submodule is a strongly coprimarily packed submodule.

Proposition
If M is a p-compactly packed module, which cannot be written as a finite union of primary submodules, then M is a coprimarily packed module.
Proof.Let N N where N is a proper submodule and N α is a primary submodule of M for each α ∈ Λ .Since M is a p-compactly packed module then there exists 1 2 , , , n α α α ∈ Λ  such that  so there exists n N ∈ and . By hypothesis

Definition
Let M be a non-zero M is called Primary Module if the zero-submodule of M is a primary submodule.

Proposition
If M is a multiplication or finitely generated strongly p-compactly packed module, then M is a strongly coprimarily packed module.The converse holds if M is a primary module such that every primary submodule of M contains no non-trivial primary submodule.
Proof.Suppose M is a primary module such that every primary submodule of M contains no non-trivial primary submodule.Let N be a proper submodule of M such that N N α α∈Λ ⊆  where N α is a primary submodule of M for each α ∈ Λ .Without loss of generality we can suppose that 0 N α ≠ , for each α ∈ Λ .Then N α is a maximal submodule of M, for each α ∈ Λ .Since M is strongly coprimarily packed module, there ex- ists β ∈ Λ , such that .Therefore M is a strongly p-compactly packed module.The other direction is trivial.

.
If there exists β ∈ Λ such that N N M β + ≠ , then N is called Strongly Coprimely Packed submodule.
for some β ∈ Λ , then N is called Strongly P-Compactly Packed.A module M is said to be P-Com- pactly Packed (Strongly P-Compactly Packed) if every proper submodule of M is p-compactly packed (strongly p-compactly packed).
hence N N β ⊆