Some Results on Weak Essential Submodules

Throughout this paper R represents commutative ring with identity and M is a unitary left R-module. The purpose of this paper is to investigate some new results (up to our knowledge) on the concept of weak essential submodules which introduced by Muna A. Ahmed, where a submodule N of an R-module M is called weak essential, if N ∩ P ≠ (0) for each nonzero semiprime submodule P of M. In this paper we rewrite this definition in another formula. Some new definitions are introduced and various properties of weak essential submodules are considered.


Introduction:
Let R be a commutative ring with identity and let M be a unitary left Rmodule.Assume that all R-modules under study contain semiprime submodules.It is well known that a submodule N of M is called essential, if whenever N ∩ L = (0), then L = (0) for each nonzero submodule L of M [1] and [2].].A proper submodule P of M is called prime, if whenever rm  P for r R and m M, then either m  P or r  ( M) [3].A nonzero submodule N of M is called semi essential, if N ∩ P ≠ (0) for each nonzero prime submodule P of M [4].This paper consists of three sections; in section 1, we give some remarks and examples, and discuss the transitivity property of weak essential submodules.In section 2, we introduce some new results (up to our knowledge) on the concept of weak essential submodules.Section 3, is devoted to the study ascending and descending chain conditions on weak essential submodules.
Muna in [5] introduced the concept of weak essential submodules as a generalization of the class of essential submodules.A proper submodule N of M is called semiprime , if for each rR and xM with r k xL, then rxL [6].Equivalently, if r 2 x  L, then rxL [7].And a submodule N of M is called weak essential, if N ∩ L ≠ (0), for each nonzero semiprime submodule L of M. Muna saw in [5] that the class of weak essential submodules lies between the class of essential submodules and the class of semi essential submodules.
In this work we give some new results (up to our knowledge) about this class of submodules.
Firstly we rewrite the definition of weak essential submodules in another formula.In fact we did not find any reasonable reason to exclude the zero submodule from this definition.We find it may be useful in some cases instead of the origin formula.Definition (1): A submodule N of an Rmodule M is called weak essential, if whenever N∩P = (0), then P = (0) for every semiprime submodule P of M.
We see that in order to add other results for weak essential submodules, it must be necessary giving some other simple remarks about this class of submodules as well as the remarks which were mentioned in [5].

Remarks (2):
1.When a submodule N of an Rmodule M is nonzero in the Def (1), then N is a weak-essential submodule if N ∩ P ≠ (0) for each semiprime submodule P of M, and this is the same definition which is mentioned in [5].

2.
Every module is a weak essential submodule in itself.

3.
In the concept of the essential submodules, (0) is an essential submodule of an R-module M if and only if M = (0).But in the concept of weak essential submodules this statement is not satisfying.In fact (0) ≤ weak (0), but sometimes (0) may be weak essential submodule in a nonzero module, for example ( ̅ ) is a weak essential submodule of the Z-module, Z 5 , and in other examples such as Zmodule Z, we note that (0) is not weak essential submodule.

5.
The sum of two weak essential submodules is also weak essential submodule.Proof (5): Let M be an R-module and let L and K be two weak essential submodules of M. Note that L ≤ L+K, since L ≤ weak M, so by [5, Rem(1.5)(2)],L+K ≤ weak M.
In the following proposition we prove the transitive property for nonzero submodules.Before that we need the following Lemma which appeared in [7,Prop (1.11), p.48].Lemma (4): If P is a semiprime submodule of C and B is a submodule of an R-module C, such that B ≰ P, then P ∩ B is a semiprime submodule in B. Proposition (5): Let C be an Rmodules, and let A, B be submodules of C such that (0) ≠A ≤ B ≤ C. If A ≤ weak B and B ≤ weak C, then A ≤ weak C. Proof: Let P be a semiprime submodule of C such that A ∩ P = (0).Note that (0) = A ∩ P = (A ∩ P) ∩ B = A ∩ (P ∩ B).But P is a semiprime submodule of C, so we have two cases.If B ≤ P then (0) = A ∩ (P ∩ B) = A ∩ B, hence A ∩ B = (0), but A ≤ B, so A ∩ B = A, which is implies that A = (0).But this is a contradiction with our assumption.Thus B ≰ P, and by Lemma (4), P ∩ B is a semiprime submodule of B. But A ≤ weak B, therefore P ∩ B = (0), and since B ≤ weak C, then P = (0), that is A ≤ weak C.

Remark (6):
The condition A ≠ (0) in Prop (1.5) is necessary.In fact in the Zmodule Z 8 , ( ) is a weak essential submodule of { , } and { , } is a weak essential submodule of Z 8 , but ( ) not weak essential in Z 8 .
The converse of Prop ( 5) is not true in general, as the following example shows.Example (7): Consider the Z-module, Z 36 , the submodule ( ) is a weak essential submodule of Z 36 .But ( ) is not weak essential submodule of ( ).

1.Other results on weak essential submodules
In this section we introduce other properties of weak essential submodules.We start by the following definition which is analogue of that in [8].

Definition (1.1):
A nonzero R-module is called fully essential*, if every nonzero weak essential submodule of M is an essential submodule of M.
It is clear that every fully essential* module is a fully essential module, since every weak essential submodule is a semi-essential submodule [5].
Recall that an R-module M is called fully semiprime, if every proper submodule of M is a semiprime submodule [9].
Before giving the following proposition, we need to introduce the following lemma.

Lemma (1.2):
Let A and B be submodules of an R-module M such that A ≤ B. If A is a semiprime submodule of M, then A is a semiprime submodule in B. Proof: It is clear.Proposition (1.3):Let M be a fully semiprime R-module, and let N ≤ M. Then N ≤ weak L if and only if N ≤ e L for every submodule L of M. Proof: ⇒) Let L be a submodule of M and let A be a submodule of L such that N ∩ A = (0), since M is a fully semiprime module then both of N and A are semiprime submodules of M, and by Lemma (1.2), N is a semiprime submodule of L. But N is a weak essential submodule of L, therefore A = (0), that is N is an essential submodule of L. ⇐) It is clear.Corollary (1.4):If M is a fully semiprime module, then every nonzero weak essential submodule of M is an essential submodule of M. Corollary (1.5): Every fully semiprime module is a fully essential* module.
Recall that a nonzero R-module M is called weak uniform if every nonzero R-submodule of M is a weak essential.A ring R is called weak uniform if R is a weak uniform R-module, [5].Proposition (1.6):Let M be an Rmodule, then M is uniform module if and only if M is weak uniform and fully essential* module.
Since M is a weak uniform module, then is a weak essential submodule of M. But M is a fully essential* module, therefore N is an essential submodule of M, and we are done.Corollary (1.7):Let M be a fully semiprime R-module, then a nonzero module M is a uniform if and if M is a weak uniform module.
The following theorem gives the hereditary of "fully essential* property" between the ring R and the module M which defined on R. Theorem (1.8):Let M be a nonzero finitely generated, faithful and multiplication R-module.Then M is a fully essential* module if and only if R is a fully essential* ring.Proof: ⇒) Assume that M is a fully essential* module, and let I be a nonzero weak essential ideal of R, then IM is a submodule of M say N. Since M is a finitely generated, faithful and multiplication module so by [5,Th (3.6)], N is a weak essential submodule of M. Since I ≠ (0) and M is a faithful module, then N ≠ (0).But M is a fully essential* module, therefore N is an essential submodule of M. Since M is a faithful and multiplication module, thus I is an essential ideal of R [10, Th (2.13)], that is R is a fully essential* ring.⇐) Suppose that R is a fully essential* ring and let (0) ≠ N ≤ weak M. Since M is a multiplication module, then there exists a weak essential ideal I of R such that N = IM [5].By assumption I is an essential ideal of R.But M is a finitely generated faithful and multiplication module, then N is an essential submodule of M [10, Th (2.13)], and we are done.
The following proposition deals with the direct sum of weak essential submodules.Proposition (1.9):Let M = M 1  M 2 be a fully semiprime R-module where M 1 and M 2 are submodules of M, and let (0) ≠ K 1 ≤ M 1 and (0) ≠ K 2 ≤ M 2 .Then K 1  K 2 is a weak essential submodule of M 1  M 2 if and only if K 1 is a weak essential submodule of M 1 and K 2 is a weak essential submodule of M 2 .Proof: ⇒) Since M is a fully semiprime module, then by Cor (1.4), K 1  K 2 is an essential submodule of M 1  M 2 , and by [11], K 1 is an essential submodule of M 1 and K 2 is an essential submodule of M 2 .But every essential submodule is a weak essential, therefore K ≤ weak M 1 .

⇐) It follows similarly
In the following proposition we give another case for the direct sum of weak essential submodules.Proposition (1.10):Let M = M 1  M 2 be an R-module where M 1 and M 2 are submodules of M, and let K 1 ≤ M 1 and K 2 ≤ M 2 .If K 1  K 2 is a weak essential submodule of M 1  M 2 , then K 1 is a weak essential submodule of M 1 , provided that every semiprime submodule of M 1 is a semiprime submodule of M. Proof: Let P 1 is a semiprime submodule of M 1 such that K 1 ∩ P 1 = (0).By using some properties in set theory, we can easily show that (K 1  K 2 ) ∩ P 1 = (0).But K 1  K 2 ≤ weak M and by assumption P 1 is a semiprime submodule of M, Thus P 1 = (0).Let us introduce the following definition.

Definition (1.11):
Let M be an Rmodule and let N be a submodule of M.
A semiprime submodule L of M is called weak-relative intersection complement of N in M, if whenever N∩P = (0), where P is a semiprime submodule of M, such that L  P, then L = P.In other words L is a maximal submodule with the property N∩L = (0).Remark (1.12):It is well known that every submodule of an R-module has a relative complement [1, P.17].We verify by example that not every submodule has a weak-relative intersection complement, for example; the submodule ( ) of Z 4 -module Z 4 hasn't weak-relative intersection complement, since there exists only one submodule ( ) of Z 4 such that ( ) ∩ ( ) = ( ), and ( ) is not semiprime submodule of Z 4 as Z 4, i.e. ( ) is not semiprime ideal of the ring Z 4 .In fact ( ) is not the only nilpotent ideal in the ring Z 4 , so by [1, P.2], ( ) is not nilpotent ideal of Z 4 .
Muna in [5] showed by example that the intersection of two weak essential submodules need not be weak essential submodule, and she satisfied that under certain condition, see [5,Prop (1.6)].In this work we give a different condition.Proposition (1.13):Let M be an Rmodule and let N 1 and N 2 be a weak essential submodules of M such that N 1 ∩ N 2 ≠ (0) and all semiprime submodules of N 1 are semiprime submodules of M, then N 1 ∩ N 2 ≤ weak M. Proof: Let P be a semiprime submodule of M such that (N 1 ∩ N 2 ) ∩ P = (0).This implies that N 2 ∩ (N 1 ∩ P) = (0).If N 1 ≤ P, then we have a contradiction with the assumption, thus N 1 ≰ P. By Lemma (1.4), N 1 ∩ P is a semiprime submodule of N 1 .Since N 2 ≤ weak M and by our assumption N 1 ∩ P is a semiprime submodule of M, we have N 1 ∩ P = (0).But N 1 ≤ weak M, therefore P = (0), hence N 1 ∩ N 2 ≤ weak M. Note: The condition "all semiprime submodules of N 1 are semiprime submodules of M" in Prop (1.13), can also be applied for N 2 .Proposition (1.14):Let M be an Rmodule and N 1 and N 2 are weak essential submodule of M such that N 2 ∩ P is a semiprime submodules of M for all semiprime submodule P of M, then N 1 ∩ N 2 ≤ weak M. Proof: Let P be a semiprime submodule of M such that (N 1 ∩ N 2 ) ∩ P = (0).This implies that N 1 ∩ (N 2 ∩ P) = (0).Since N 2 ∩ P is a semiprime submodule of M and N 1 ≤ weak M, then N 2 ∩ P = (0).But N 2 ≤ weak M, thus P = (0).
As a generalization of the result in [11, Prop (5.21), P.75], we give the following proposition.Proposition (1.15):Let N be a nonzero R-module of M, and let N' be a nonzero semiprime submodule of M. If N' is a weak relative intersection complement of N in M, then is a weak essential submodule of .
Proof: Let g: M → be a natural epimorphism, and let N' be a weak relative complement of N in M. Let be a nonzero semiprime submodule of such that ∩ = (0).By [7], g - 1 ( ) is a semiprime submodule of M [5, P.216], put g -1 ( ) = P for some semiprime submodule P of M, then g(P)

=
. Thus ∩ = (0), this implies that ( ) = (0) hence (N N') ∩ K = N'.By modular law N ∩K  N', that is N ∩ K  N' ∩ N. Since N' is a weak relative complement of N in M, then N' is the maximal submodule with the property N ∩ N' = (0).It follows that N ∩ K = (0), and by maximality of N' we get K = N', therefore = (0).That is is a weak essential submodule of .We need the following definition which appeared in [12].Definition (1.16):Let M be an Rmodule and N ≤ M. If there exists a semiprime submodule of M containing N, then the intersection of all semiprime submodule of M containing N is called semi-radical of N, and it is denoted by S-rad N. If there is no semiprime submodule of M containing N, then we say that S-rad N = M, in particular Srad M = M. Proposition (1.17):Let M be an Rmodule and let (0) ≠ N ≤ M. If N' is a weak relative complement of N in M, and N' ≤ S-rad(M), then N  N' ≤ weak M.
Hence N  N' ≤ weak M.
Recall that an R-module M is called multiplication, if for each submodule N of M, there exists an ideal I of R such that N=IM [13].Proposition (1.18):Let M be a faithful and multiplication module such that M satisfies the condition (*), and let I, J be ideals of R. If IM ≤ weak JM, then I ≤ weak J, where: Condition (*): For any two ideals L and K of R, if L is a semiprime ideal of K, then LM is a semiprime submodule of KM.Proof: Let P be a semiprime ideal of J such that I ∩ P = (0), then IM ∩ PM = (0). .Since M is a faithful and multiplication, therefore IM ∩ PM = (0) [10,Th (1.7)].By condition (*), PM is a semiprime submodule of JM.But IM ≤ weak JM, then PM = (0).Since M is a faithful module so P = (0), thus I ≤ weak J.Note that the condition (*) which mentioned in Prop (1.18) is not hold in general, as shown in the following example.

Example (1.19):
The Z 4 -module, Z 4 is not satisfying the condition (*), since there exists a prime ideal I = { , } of the ring Z 4 , with IZ 4 not prime submodule of Z 4 .In fact IM = {∑ The converse of Prop (1.18) is true without using the condition (*), but we need to add another condition as the following proposition shows.Proposition (1.20):Let M be a finitely generated, faithful and multiplication Rmodule.If I ≤ weak J then IM ≤ weak JM for every ideals I and J of R. Proof: Let P be a semiprime submodule of JM such that IM ∩ P (0).Since M is a multiplication faithful module, then P = EM for some semiprime ideal E of R [14, Prop (2.5), P.36].So IM ∩ EM = (0), this implies that (I ∩ E) M = (0).Since M is a faithful module, then I ∩ E = (0).On the other hand since EM ≤ JM and M is a finitely generated, faithful and multiplication module so by [10,Th (3.1)] E ≤ J.But E is a semiprime ideal of R, then by Lemma (1.2), E is a semiprime ideal of J. Since I is a weak essential ideal of J, then E = (0), and hence P = (0).That is IM ≤ weak JM.
From Prop (1.18) and Prop (1.20) we have the following theorem.Theorem (1.21):Let M be a finitely generated, faithful and multiplication module such that M satisfies the condition (*).Then I ≤ weak J if and only if IM ≤ weak JM for every two ideals I and J of R.
It is well known that If a ring R has only one maximal ideal I, then I is an essential ideal of R if and only if I ≠ (0).In the following proposition we generalize one direction of this statement for essential (

2.Modules with ACC (DCC) on weak essential submodules
Recall that an R-module M called satisfies ACC (DCC), if each ascending (descending) condition of submodules of M is finite [2].In this section we study this property on a special class of submodules which is the class of weak essential submodules.We study the hereditary property for this definition between M and it's submodules, and between M and the ring R which defined on it.We start by the following definition.Definition (2.1):An R-module M is called satisfied the ascending chain condition (ACC) on weak essential submodules if each ascending chain of weak essential submodules N 1  N2  ...  Nn  ...is finite.And M is called satisfied descending chain condition (DCC) on weak essential submodules if each descending chain of weak essential submodules N 1  N2  ...  Nn  … is finite.
[8,8, Prop (1.6), P. 7], if an R-module M is finitely generated, then every proper submodule of M is contained in a maximal submodule of M. If we use this statement and replace the condition "nonzero multiplication module" in Prop (1.22) by the condition "finitely generated module", then we get the same result.