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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 where N_{a} is a primary submodule of M for each , then for some . When there exists such that ; 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.

Coprimely packed rings were introduced by Erdo˘gdu for the first time in [

In this paper, we discuss the situation where the union of a family of primary submodules of M is considered.

In [

P-Compactly Packed. If for each family

ist

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.

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

Let N be a proper submodule of an R-module M. N is said to be Coprimarily Packed Submodule if whenever

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

Proof. Suppose that M is a strongly coprimarily packed module and let

We must show

Suppose

Thus

this implies

which is a contradiction. So

Conversely, suppose

and since

Suppose

The following proposition gives a characterization of strongly coprimarily packed submodules in a multiplication or finitely generated module.

Let M be a finitely generated or multiplication R-module. A proper submodule N is strongly coprimarily packed if and only if whenever

Proof. Suppose N is a strongly coprimarily packed submodule and let

Conversely, let

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.

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

First, if

Now, if

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

of primary submodules of M with

It is easy to show that every strongly p-compactly packed submodule is a strongly coprimarily packed submodule.

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

We claim that

By hypothesis

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

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

The other direction is trivial.

Lamis J. M.Abulebda， (2015) On Co-Primarily Packed Modules。 Advances in Pure Mathematics，05，208-211. doi: 10.4236/apm.2015.54022