1. Introduction
The prime radical of a submodule
of an R-module
, denoted by
is defined as the intersection of all prime submodules of
which contain
, if there exists no prime submodule of
containing
, we put
[1].
We naturally seek a counterpart in the primary radical of a submodule of module.
Firstly we introduced a definition for the primary radical of a submodule with some of its basic properties. We also define the P-radical submodule and review some results about it.
Finally, we find a method to characterize the primary radical of a finitely generated submodule of a free module.
2. Some Basic Properties of the Primary Radical
In this section we introduce the concept of the primary radical and give some useful properties about it.
2.1. Definition
The primary radical of a submodule
of an R-module
, denoted by
is defined as the intersection of all primary submodules of
which contain
. If there exists no primary submodule of
containing
, we put
.
If
, since the primary submodules and the primary ideals are the same, so if
is an ideal of
,
is the intersection of all primary ideals of
, which contain
. Now, we give useful properties of the primary radical of a submodule.
2.2. Proposition
Let
and
be submodules of an R-module
. Then 1) ![](https://www.scirp.org/html/9-5300208\fafdd7a5-27e9-49a7-a8fc-8c327125d8c9.jpg)
2) ![](https://www.scirp.org/html/9-5300208\6ac7135e-8815-442e-9f22-2a28346aa74f.jpg)
3) ![](https://www.scirp.org/html/9-5300208\2ad913b9-ce1e-4151-b9a5-7eec20b4de69.jpg)
Proof.
1) It is clear.
2) Let
be primary submodule of
containing L,
since
so
. Thus
.
By the same way
.
It follows
.
3) By 1) we have
. Now
where the intersection is over all primary submodules
of
with
.
![](https://www.scirp.org/html/9-5300208\7e4b6b3a-a218-4649-95c8-26b4b6b994f6.jpg)
In the following two propositions, we give a condition under which the other inclusion of 2) holds, that is;
provided that every primary submodule of
which contains
is completely irreducible submodule. Where a submodule
of an
-module
is called Completely Irreducible if whenever
, then either
or
where
and
are submodules of
.
2.3. Proposition
Let
and
be submodules of an
-module
. If every primary submodule of
which contains
is completely irreducible submodule, then:
.
Proof. By proposition (2.2, (2))
. If
, clearly
. If
, there exists a primary submodule
of
such that,
by hypothesis either
or
so that either
or
, because every primary submodule containing
, so either
or
therefore
.
2.4. Proposition
Let
and
be submodules of an
-module ![](https://www.scirp.org/html/9-5300208\fee4f816-e2dc-48c3-a44f-f347c3c0e547.jpg)
such that
, then
.
Proof. If
is a primary submodule containing
, then
. So
.
Since
is a prime ideal, either
or
.
If
,
then
for otherwise
which is a contradiction. Therefore
. Now, applying proposition (2.3), we can conclude that
.
We conclude the same result if
.
Let
be a proper submodule of an R-module
. Let
be a prime ideal of R. For each positive integer
, we shall denote by
the following subset of
![](https://www.scirp.org/html/9-5300208\625a884d-64fa-44f2-b1a8-f91be48e9abe.jpg)
2.5. Proposition
Let
be a submodules of an R-module
and
be a prime ideal of R. For each positive integer
:
or
is a
-primary submodule of
.
Proof. Let
be any positive integer, it is clear that
is a submodule of
.
Assume
. To show
is
-primary,
that is
. Nowlet
be a submodule of
properly containing
, let
,
.
Since
, let
, but
thus
, there exists
such that
. If
, then
and this implies
, which is a contradiction. It follows
, therefore
.
So
is a primary submodule
, we have proved above that
, that is
.
Let
,
for some
, thus
for some
. If
then
this implies
, which is a contradiction. Therefore
thus
.
The following theorem gives a description of the primary radical of a submodule.
2.6. Theorem
Let
be a submodule of a module
over a Noetherian ring
. Then
![](https://www.scirp.org/html/9-5300208\a6f28dab-d4d8-468d-97dd-9aa502a4c989.jpg)
Proof. By proposition (2.2), for each positive integer
and any prime ideal
we have
is a
-primary submodule containing
. Hence
![](https://www.scirp.org/html/9-5300208\d4572949-ddf7-4a0b-88d6-884052abd97c.jpg)
For every primary submodule
containing
with
there exists a positive integer
such that
. So
![](https://www.scirp.org/html/9-5300208\b435f4ed-17d7-4290-b2f6-cb065cdf8c70.jpg)
Thus
![](https://www.scirp.org/html/9-5300208\f80827a5-5c59-4ba2-9ffe-808f5fe414c8.jpg)
We will give the following definition.
2.7. Definition
A proper submodule
of an R-module
with
will be called P-Radical Submodule.
Now, we are ready to consider the relationships among the following three statements for any r-module
.
1)
satisfies the ascending chain condition for pradical submodules.
2) Each p-radical submodule is an intersection of a finite number of primary submodules 3) Every p-radical submodule is the p-radical of a finitely generated submodule of it.
2.8. Proposition
Let
be an
-module. If
satisfies the ascending chain condition for p-radical submodule of
is an intersection of a finite number of primary submoules.
Proof. Let
be a p-radical submodule of
and put
, where
is a primary submodule for each
, and the expression is reduced. Assume that
is an infinite index set. Without loss of generality we may assume that
is countable, then
is an ascending chain of p-radical submodules, since by proposition (2.2),
![](https://www.scirp.org/html/9-5300208\1f870d1a-4dd4-4a18-b146-9fe2d480c099.jpg)
By hypothesis this ascending chain must terminate, so there exists
such that
, whence
which contradicts that the expression
is a reduced. Therefore
must be finite.
2.9. Proposition
Let
be an r-module. If
satisfies the ascending chain condition for p-radical submodules, then every p-radical submodule is the p-radical of finitely generated submodule of it.
Proof. Assume that there exists a p-radical submodule
of
which is not the p-radical of a finitely generated submodule of it. Let
and let
so
, hence there exists
. Let
, then
, thus there exists
, etc. This implies an ascending chain of p-radical submodules,
which does not terminate and this contradicts the hypothesis.
2.10. Proposition
Let
be a finitely generated r-module. If every primary submodule of
is the p-radical of a finitely generated submodule of it, then
satisfies the ascending chain condition for primary submodules.
Proof. Let
be an ascending chain of primary submodules of
. Since
is finitely generated then,
is a primary submodule of
.
Thus by hypothesis,
is the p-radical for some finitely generated submodule
, hence
, then there exists
such that
hence
. Thus
for some
. Therefore the chain of primary submodules
terminates
3. The Primary Radical of Submodules of Free Modules
In this section we describe the elements of
, where
is a finitely generated submodule of the free module
. Let
be a positive integer and let
be the free
-module
.
Let
for some
, then
,
, for some
,
,
.
We set
![](https://www.scirp.org/html/9-5300208\60e5b49f-dd3a-4097-b078-9503aa348fcd.jpg)
Thus the jth row of the matrix
consists of the components of the element
in
. Let
.
By a
minor of
we mean the determinant of a
submatrix of
, that is a determinant of the form:
![](https://www.scirp.org/html/9-5300208\03b3690a-39a6-4c28-a05f-c41fac83e629.jpg)
where
,
. For each
.
We denote by
the ideal of
generated by the
minors of
.
Note that
, where
.
The key to the desired result is the following two propositions.
3.1. Proposition
Let
be a ring and
be the free
-module
, for some positive integer
. Let
be a finitely generated submodule of
where
. If
, then
in
![](https://www.scirp.org/html/9-5300208\0ca70791-33f5-4802-8167-df090c778fd1.jpg)
Proof. Suppose
where
,
. Let
be any maximal ideal of
and
such that
. By proposition (2), there exists
,
,
and
such that
, where
, that is, if
where
, then 3.1) ![](https://www.scirp.org/html/9-5300208\14d2a7f0-4649-44b2-be3e-622dc81cb5c9.jpg)
Suppose that
,
![](https://www.scirp.org/html/9-5300208\dfe7920b-f3eb-4bf6-b399-78a13a2212e4.jpg)
Let
![](https://www.scirp.org/html/9-5300208\26df9f0e-5b25-4cd1-b3f9-a89745e71d8b.jpg)
which is a
minor of
. Then by (3.1)
![](https://www.scirp.org/html/9-5300208\e239c786-0132-4f41-a20c-8008d54a25f9.jpg)
which is primary with
(note that, here
) hence
. It follows
for every maximal ideal
with
for some
and
.
3.2. Proposition
Let
be a ring and
be the free
-module
for some positive integer
. Let
be a finitely generated submodule of
where
. If
in
![](https://www.scirp.org/html/9-5300208\b1561cf8-3016-4ec8-82c7-fd501002c79c.jpg)
, then
.
Proof. Suppose
![](https://www.scirp.org/html/9-5300208\3568e991-f492-4c03-9018-82fadc136df1.jpg)
and
. Let
be any prime ideal of
and
any positive integer. It is enough to show that
for all
.
If
, then
, hence
Suppose SSS
.
Note that ![](https://www.scirp.org/html/9-5300208\c77ac79e-c207-4306-bce4-0f2e8adf73ee.jpg)
Thus there exists
such that
but
is a subset of
, there exists
, such that
![](https://www.scirp.org/html/9-5300208\bdcf467a-85c5-4fef-8a08-b3d069229fc4.jpg)
By hypothesis, for each ![](https://www.scirp.org/html/9-5300208\c4377c48-94fd-4f50-a351-5cd18ca10462.jpg)
![](https://www.scirp.org/html/9-5300208\7acd523f-1ca5-47e8-bb82-dbf3a9806aa9.jpg)
Expanding this determinant by first column we find that
where
![](https://www.scirp.org/html/9-5300208\5a2472f3-1949-43c0-a9cd-f6e634e302e0.jpg)
For each ![](https://www.scirp.org/html/9-5300208\8f09e216-7511-4edb-90df-8cf6963fdafd.jpg)
Note that
and
are independent of
. Thus
![](https://www.scirp.org/html/9-5300208\6ce86892-b1c0-4501-8ba0-8e78fdc6f701.jpg)
i.e.
with
, hence
.Thus
.
3.3. Proposition
Let
and
be
-modules and
![](https://www.scirp.org/html/9-5300208\7a9e6911-aeb7-456f-a0a1-5164959b5ca4.jpg)
Let
be a proper submodule of
,
then
if and only if
.
Proof: Suppose first that
. Let
be any primary submodule of
such that
. Let
.
is a submodule of
and if
then
is a primary submodule of
since, if
where
and
, then
, which is primary submodule of
, hence either
thus
or
for some
that is,
, so
, therefore
, thus
for some
, that is
. Hence
is a primary submodule of
containing
. Thus
, so
. It follows
Conversely, suppose that
. Let
be a primary submodule of
such that
. Then
is a primary submodule of
containing
. Hence
, that is ![](https://www.scirp.org/html/9-5300208\9bdb63bb-094c-42b3-b153-21f41384b9f1.jpg)
Now, we have the main result of this section.
3.4. Theorem
Let
be a ring and
be the free
-module
, for some positive integer
. Let
be a finitely generated submodule of
where
. If
, then
in
![](https://www.scirp.org/html/9-5300208\8f27b988-81a9-4595-9490-4bfbf1808d13.jpg)
.
Proof. Let
. Suppose first
, that is
, by proposition (3.1), if
, then
in
![](https://www.scirp.org/html/9-5300208\c663a3cf-3c3c-4374-b382-4ae6cf286d47.jpg)
.
Now suppose
i.e.
. Let
for some
and
,
. By proposition (3.3),
if and only if
in
. Where
![](https://www.scirp.org/html/9-5300208\f8747359-9095-4c77-ba1a-aeae6a49be85.jpg)
Now apply proposition (3.1) to obtain the result.
The following example will illustrate application of the proposition (3.2).
3.5. Example
Let
and
be the submodule
of
. Then
if
in 2Z and
.