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In this paper 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. We find a method to characterize the primary radical of a finitely generated submodule of a free module.

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 [

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.

In this section we introduce the concept of the primary radical and give some useful properties about it.

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.

Let and be submodules of an R-module. Then 1)

2)

3)

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.

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.

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 .

Let and be submodules of an -module

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

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.

Let be a submodule of a module over a Noetherian ring. Then

Proof. By proposition (2.2), for each positive integer and any prime ideal we have is a -primary submodule containing. Hence

For every primary submodule containing with

there exists a positive integer such that. So

Thus

We will give the following 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.

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),

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.

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.

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

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

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:

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.

Let be a ring and be the free -module, for some positive integer. Let be a finitely generated submodule of where. If, then in

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)

Suppose that

,

Let

which is a minor of. Then by (3.1)

which is primary with (note that, here

) hence. It follows for every maximal ideal with for some and

.

Let be a ring and be the free -modulefor some positive integer. Let be a finitely generated submodule of where. If in

, then.

Proof. Suppose

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

Thus there exists such that but is a subset of, there exists , such that

By hypothesis, for each

Expanding this determinant by first column we find that where

For each

Note that and are independent of. Thus

i.e. with , hence.Thus.

Let and be -modules and

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

Now, we have the main result of this section.

Let be a ring and be the free -module, for some positive integer. Let be a finitely generated submodule of where. If, then in

.

Proof. Let. Suppose first, that is, by proposition (3.1), if, then in

.

Now suppose i.e.. Let for some and,. By proposition (3.3), if and only if in. Where

Now apply proposition (3.1) to obtain the result.

The following example will illustrate application of the proposition (3.2).

Let and be the submodule

of. Then

if in 2Z and.