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 Na 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