Let N be a proper submodule of an R-module M. N is said to be Coprimarily Packed Submodule if whenever where is a primary submodule of M for each, then for some. When there exists such that, N is called Strongly Coprimarily Packed submodule.

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 be an epimorphism. If M is an R-module such that for every primary submodule N of M, then M is a strongly coprimarily packed module if and only if is a strongly coprimarily packed module.

Proof. Suppose that M is a strongly coprimarily packed module and let where is a proper submodule of and is a primary submodule of for each, so

.

is a primary submodule of M for each there exists such that

.

We must show.

Suppose, let so there exists and such that. Since f is an epimorphism there exists such that

, , and.

Thus, so, this implies. Since,

, and, so, hence

,

this implies

which is a contradiction. So is a strongly coprimarily packed module.

Conversely, suppose is a strongly coprimarily packed module and let, where N is a proper submodule of M and is a primary submodule of M for each. Hence

and since for each, is a primary submodule of, there exists such that.

Suppose and let, since f is an epimorphism, there exists such that and there exists and such that. Then, hence, so). It follows which is a contradiction, thus, 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.

Let M be a finitely generated or multiplication R-module. A proper submodule N is strongly coprimarily packed if and only if whenever where is a maximal submodule of M for each then there exists such that.

Proof. Suppose N is a strongly coprimarily packed submodule and let where is a maximal submodule of M for each, hence is a primary submodule, so there exists such that. But and is a maximal submodule, hence thus.

Conversely, let where is a primary submodule for each. There exists a maximal submodule that contains for each, hence. By hypothesis, there ex- ists such that, but so, 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.

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 where is a maximal submodule of M for each, then by proposition (2.3), it is enough to show that there exists such that.

First, if, 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 there exists such that hence.

Now, if, let and, so is an -closed subset of M. Since, so thus there exists a submodule K that contains N and K is a maximal in [1] , K is prime [1] , so it is primary submodule. Thus by hypothesis K is strongly coprimarily packed and since and by proposition (2.3) there exists such that.

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, there exist such that. If

for some, 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).

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 where N is a proper submodule and is a primary submodule of M for each. Since M is a p-compactly packed module then there exists such that.

We claim that, let so there exists and such that. Then there exists such that, hence, so, thus

.

By hypothesis therefore.

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 where is a primary submodule of M for each. Without loss of generality we can suppose that, for each. Then is a maximal submodule of M, for each. Since M is strongly coprimarily packed module, there exists, such that; but is a maximal submodule and. This implies, and hence. Therefore M is a strongly p-compactly packed module.

The other direction is trivial.