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FBN ModulesJaime Castro Pérez a & José Ríos Montes ba Departamento de Matemáticas, Instituto Tecnológico y de Estudios Superiores deMonterrey, Tlalpan, México, D.F., Méxicob Instituto de Matemáticas, Universidad Nacional Autónoma de México, Area de laInvestigación Científica, México, D.F., México

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To cite this article: Jaime Castro Pérez & José Ríos Montes (2012): FBN Modules, Communications in Algebra, 40:12,4604-4616

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Communications in Algebra®, 40: 4604–4616, 2012Copyright © Taylor & Francis Group, LLCISSN: 0092-7872 print/1532-4125 onlineDOI: 10.1080/00927872.2011.613879

FBN MODULES

Jaime Castro Pérez1 and José Ríos Montes21Departamento de Matemáticas, Instituto Tecnológico y de EstudiosSuperiores de Monterrey, Tlalpan, México, D.F., México2Instituto de Matemáticas, Universidad Nacional Autónoma de México,Area de la Investigación Científica, México, D.F., México

For M ∈ R-Mod and � ∈ M-tors, we define the concept of fully �-bounded module as ageneralization of the concept of fully �-bounded ring. We prove that for a �-noetherianmodule M with local �M -Gabriel correspondence, which is a progenerator of ��M�

and with � is FIS-invariant, then M is fully �-bounded. Also, we show that if M is�-noetherian and fully �-bounded, then M has local �M -Gabriel correspondence.

Key Words: Ideal invariant torsion theory; M-ideal; Prime submodules.

2010 Mathematics Subject Classification: 16S90; 16D50; 16P50; 16P70.

INTRODUCTION

Left fully bounded noetherian rings have been studied by several authors.In [14], the author showed that left fully bounded noetherian rings are characterizedas those rings that have bijective Gabriel correspondence (that is, the map �E� →ass �E� from the set of isomorphism classes �E� of injective indecomposable leftR-modules to the set Spec �R� of prime ideals, is a bijection). Later, in [4] theauthor gave another characterization of fully bounded noetherian rings in terms ofhereditary torsion theories.

Later, a more general situation was considered. Let R be a ring notherian withrespect to an hereditary torsion theory �. For this kind of ring, in [2, 3, 12, 13],the authors have studied the condition of when two �-torsion free indecomposableinjective left R-modules are isomorphic if and only if they have the same associatedprime ideal. This condition is named the local bijective Gabriel correspondence withrespect to �.

An element � ∈ R-tors is an ideal invariant torsion theory if I/ID is �-torsionfor every two-sided ideal I and every left ideal D such that R/D is �-torsion. In[12, 13], the authors proved that if R is �-noetherian and � is an ideal invarianttorsion theory, then R has bijective local Gabriel correspondence with respect to � if

Received May 24, 2011; Revised July 28, 2011. Communicated by T. Albu.Address correspondence to Prof. Jaime Castro Pérez, Departamento de Matemáticas,

ITESM.CCM, Calle del Puente 222, Col Ejidos de Huipulco, México 14380, México; E-mail:jcastrop@itesm.mx

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and only if R is fully �-bounded (that is, every �-pure essential left ideal of a primefactor ring of R contains a non-zero two-sided ideal). In [1], the authors obtainedequivalent conditions to R having local bijective Gabriel correspondence relative to� when R has �-Krull dimension and Ass �M� �= ∅ for any non-zero �-torsion freeleft R module M . In [7], the authors found neccesary and sufficient conditions for aring with �−Gabriel dimension to have local Gabriel correspondence.

It is possible to work in an even more general situation. Let M be anR-module, and let � �M� denote the full subcategory of R-Mod whose objectsare R-modules isomorphic to submodules of M-generated modules. In [5], theconcept of prime M-ideal for M ∈ R-mod is defined, and the concept of Gabrielcorrespondence is extending to M-injective modules. This is made by showing thatif M is a left noetherian module satisfying condition H (see [5], Definition 6.5)and such that Hom �M�X� �= 0 for all modules 0 �= X ∈ � �M�, then there is abijective correspondence between the set of isomorphism classes of indecomposableM-injective modules in � �M� and the set of prime M-ideals.

In a very recent article from [9], the authors use the concept of primesubmodule defined in Raggi et al. [15, Definition 13]), to find necessary and sufficientconditions for a projective module M in � �M�, which generates the category � �M�,and with �M -Gabriel dimension, to have a one-to-one correspondence between theset of representatives the of isomorphism classes of indecomposable �-torsion freeinjective modules in � �M� and the set of �-pure submodules prime in M .

In this article, we continue the investigation started in the article “Primesubmodules and local Gabriel correspondence in ��M�” [9]. In particular, we define,for an R-module M and � ∈ M-tors, the concepts of �-bounded module and fully �-bounded module. Also, we define the concept FIS-invariant for a � ∈ M-tors andwe prove that if M is a �-noetherian, projective in � �M�, and a generator of thecategory � �M� with � FIS-invariant, then, the following conditions are equivalent:

(1) M has �M -Gabriel correspondence,(2) M is fully �-bounded.

As a consequence, we obtain that if M satisfies the previous conditions, thenthe following are equivalent:

(1) M is fully �-bounded,(2) M has �-�M -dimension.

In order to do this, we organized the article in two sections. In Section 1, wegive some results about the category � �M� for M projective in � �M�. In particular,we show (Proposition 1.3) that if M is a prime R-module, M being �-torsion-free,�-noetherian and projective in � �M�, then M is non M-singular. In Section 2, wedefine the concept of being FIS-invariant for � ∈ M-tors and we define the conceptof being fully �-bounded for an R-module M . In Theorem 2.8, we show that if � isFIS-invariant, M is �-noetherian and M has �M -Gabriel correspondence, then M isfully �-bounded. In Proposition 2.9, we obtain the inverse to Theorem 2.8 withoutusing the condition: � is FIS-invariant.

In what follows, R will denote an associative ring with unity and R-Modwill denote the category of unitary left R-modules. An R-module X is said to beM-generated if there exists an R-epimorphism from a direct sum of copies of M onto

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X. The category � �M� is defined as the full subcategory of R-Mod containing allR-modules X, which are isomorphic to a submodule of an M-generated module.

Let M-tors be the frame of all hereditary torsion theories on � �M�. For afamily �M of left R-modules in � �M�, let � ��M� be the greatest element of M-torsfor which all the M are torsion free, and let � ��M� denote the least elementof M-tors for which all the M are torsion. � ��M� is called the torsion theorycogenerated by the family �M� and � ��M� is the torsion theory generated by thefamily �M In particular, the greatest element of M-tors is denoted by � and theleast element of M-tors is denoted by � If � is an element of M-tors, gen ��� denotesthe interval ��� ��. Let 0 �= N ∈ ��M� and � ∈ M-tors, N is said to be �-�M -module ifN ∈ �� and � ∨ ��N� is atom in gen���.

Let � ∈ M-tors. By ������ t�, we denote the torsion class, the torsion free class,and the torsion functor associated to �, respectively. For N ∈ � �M�, N is called �-cocritical if N ∈ �� and for all 0 �= N ′ ⊆ N� N/N ′ ∈ ��. We say that N is cocriticalif N is �-cocritical for some � ∈ M-tors. For N ∈ � �M�, let N denote the injectivehull of N in � �M�. If N is an essential submodule of M , we write N ⊆ess M . For� ∈ M-tors and M ′ ∈ � �M�, a submodule N of M ′ is �-pure in M ′ if M ′/N ∈ ��.

A module N ∈ � �M� is called singular in � �M�, or M-singular, if there is anexact sequence in � �M� 0 → K → L → N → 0, with K essential in L. The class � ofall M-singular modules in � �M� is closed by taking submodules, factor modules, anddirect sums. Therefore, any L ∈ � �M� has a largest M-singular submodule � �L� =∑

�f �N� �N ∈ � and f ∈ Hom �N�L�. L is called non M-singular if � �L� = 0.Let M ∈ R-Mod. In Beachy [5, Definition 1.1]) the annihilator in M ∈ R-

Mod of a class � of R-modules is defined as AnnM��� = ⋂K∈� K, where � =

�K ⊆ M � there exists W ∈ � and f ∈ Hom�M�W� with K = ker f. Also in Beachy[5, Definition 1.5]) a product is defined in the following way. Let N be asubmodule of M . For each module X ∈ R-Mod, N · X = AnnX���, where � is theclass of modules W , such that f�N� = 0 for all f ∈ Hom�M�W�. For M ∈ R-Mod

and K, L submodules of M , in [6] is defined as the product KML as KML =∑�f �K� � f ∈ Hom �M�L�.

For details about �M -Gabriel dimension and �-PM -dimension, see [9]. For allthe other concepts and terminology concerning torsion theories in � �M�, see [19, 20].For torsion theoretic dimensions, the reader is referred to [7, 8, 10, 17].

1. PRELIMINARIES

Definition 1.1. Let M ∈ R-Mod and � ∈ M-tors. A module N ∈ � �M� is �-noetherian if the lattice of all �-pure submodules of N satisfies the ascending chaincondition.

Notice that if M is �-noetherian, then M has �′M -Gabriel dimension for all �′ ∈M-tors such that � ≤ �′ < �. In fact, let � ∈ M-tors with �′ ≤ � < �. Then M ��.So t� �M� is a proper �-pure submodule of M . Since M is �-noetherian, there exists asubmodule N of M such that N is a maximal �-pure submodule. Therefore, M/N isa �-cocritical module. So by Castro and Ríos [9, Proposition 4.1] M has �′M -Gabrieldimension.

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Proposition 1.2. Let M ∈ R-Mod. Suppose that M is a projective module in � �M�.If N ⊆ M is an M-singular submodule, then ker f ⊆ess M for all f ∈ Hom �M�N�.

Proof. Let f � M → N be a morphism. As N is M-singular, then there is an exactsequence in � �M�, 0 → K

i→ L�→ N → 0 such that K ⊆ess L. Since M is projective

in � �M�, then there exists a morphism f � M → L such that the following diagramcommutes.

So � � f = f . As K ⊆ess L, then f−1 �K� ⊆ess M . We claim that ker f =f−1 �K�. In fact, f

(f−1 �K�

)= � � f

(f−1 �K�

)= �

(f(f−1 �K�

))⊆ � �K� = 0.

Therefore, f−1 �K� ⊆ ker f . Now let m ∈ ker f , then 0 = f �m� = � � f �m� =�(f �m�

). So f �m� ∈ K. Hence, m ∈ f−1 �K�. Then ker f ⊆ f−1 �K�. So ker f =

f−1 �K�. �

From Raggi et al. [15, Definition 11 and Definition 16], we know that for M ∈R-Mod and N �= M a fully invariant submodule of M . N is prime in M if for anyK, L fully invariant submodule of M we have that KML ⊆ N implies that K ⊆ N orL ⊆ N . We say that M is a prime module if 0 is prime in M .

Proposition 1.3. Let � ∈ M-tors. Suppose that M is �-torsion-free, �-noetherian andprojective in � �M�. If M is a prime module, then M is non M-singular.

Proof. Let � �M� = Z the M-singular submodule of M in � �M�. Suppose thatZ �= 0. As M is a prime module, then by [9, Proposition 1.11] ZMZ �= 0. Hence,there is a non-zero morphism j � M → Z. Thus, ZMIm j �= 0. Now we consider thefamily � = �ker h �h ∈ Hom �M�Z� and ZMIm h �= 0. Then ker j ∈ � . So � �= ∅.On the other hand, since Z ⊆ M ∈ ��, then ker h is a �-pure submodule of M forall h ∈ Hom �M�Z�. As M is a �-noetherian module, then the family � has maximalelements. Let f ∈ Hom �M�Z� such that ker f is a maximal element of � . Now letg ∈ Hom �M�Z� and consider the composition g � f � M → Z. Let K = ker f , thenK ⊆ ker g � f . On the other hand, we know by Proposition 1.2 that ker g ⊆ess M .As 0 �= Im f ⊆ Z ⊆ M , then ker g ∩ Im f �= 0. So there exists m ∈ M such that 0 �=f �m� ∈ ker g. Hence, m ker f . Thus, K � ker g � f . As K is maximal in � and g �f ∈ Hom �M�Z�, then ZMIm g � f = 0. Since M is a prime module, then by Castro etal. [9, Proposition 1.11] we have that Im g � f = 0. Therefore g �f �M�� = 0. Hencef �M�M Z = 0. Thus by Castro et al. [9, Proposition 1.11] f �M� = 0 or Z = 0, acontradiction. Therefore, M is non M-singular.

Lemma 1.4. Let be M projective in � �M� and let P be a submodule of M . Let N ∈� �M� be such that PMN = 0, then PM �N/K� = 0 for all submodules K of N .

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Proof. Let K ⊆ N and f ∈ Hom �M�N/K�. As M is projective in � �M�, then thereexists a morphism g � M → N such that the following diagram commutes

So � � g = f . Inasmuch as PMN = 0, then g �P� = 0.So f �P� = �� � g� �P� = � �g �P�� = 0. Therefore, PM �N/K� = 0.

Proposition 1.5. Suppose that M is projective in � �M� and generator of the category� �M� . If P is a fully invariant submodule of M , then the following conditions hold.

1. � �M/P� = �N ∈ � �M� �P ⊆ AnnM �N�.2. M/P is generator of the category � �M/P�.3. M/P is projective in � �M/P�.

Proof. (1) Let L ∈ � �M/P�, then L is isomorphic to a submodule of ahomomorphic image of a direct sum of copies of M/P. As M is projective in � �M�and P as is a fully invariant submodule of M , then by Castro et al. [9, Proposition1.8] we have that PM �M/P� = 0. Thus by Castro et al. [9, Proposition 1.3] we knowthat PM �

⊕�M/P�� = 0 for every direct sum of copies of M/P. Now by Lemma 1.4,

PM

(⊕�M/P�

K

)= 0 for all submodules K ⊆ ⊕

�M/P�. Therefore, PML = 0.Now let N ∈ � �M� such that P ⊆ AnnM �N�. Then f �P� = 0 for all f ∈

Hom �M�N�. On the other hand, since M is a generator of the category � �M�,then there exists an epimorphism F � M�X� → N for some set X. We consider thefollowing diagram:

where ix is the inclusion morphism in the coordinate “x” and i is the inclusionmorphism of P in M . So we have that F � ix ∈ Hom �M�N�. Hence, �F � ix� �P� =0 for all x ∈ X. Thus, P�X� ⊆ ker F . Then we can define the morphism F �M�X�

P�X� → N such that F �a� = F �a�. As F is an epimorphism, then also F is anepimorphism. Since M�X�/P�X� �M/P��X�, then N ∈ � �M/P�. Thus, � �M/P� =�N ∈ � �M� �P ⊆ AnnM �N�. Moreover, we have shown that M/P is a generator ofthe category � �M/P�. Thus, we have (2).

(3) It was proved in Van den Berg and Wisbauer [18, Lemma 9(1)]. �

Note that in (1) the equality is not true in general. Consider the followingexample:

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Example 1.6. Let R = and let p ∈ R be a prime number. If M = , �p� = � ab∈

�p � b, Y = � apn

∈ � n ∈ �, then, M�p� = 0 and MY = 0. Hence �p�, Y ∈�N ∈ � �M� �M ⊆ AnnM �N�. Therefore, � �M/M� �= �N ∈ � �M� �M ⊆ AnnM �N�.

Example 1.7. Let R = , and let p ∈ R be a prime number. Let us take M = p�

and X = p. As p is a fully invariant submodule of p� , we consider the category� �M/X� = �

[p�

]. We have that

(p

)p�

p� = p.

Hence, p � AnnM

(p�

). Thus, p� {

N ∈ �[p�

] �p ⊆ AnnM �N�}.

Therefore, � �M/X� �= {N ∈ �

[p�

] �X ⊆ AnnM �N�}. Moreover, we also note that

in this example M/X = p� is not a generator of the category � �M/X� = �[p�

].

Hence, condition (2) of Proposition 1.5 is in general not true.The following example shows that the condition of M/P being projective in

� �M/P� does not imply that M projective in � �M�.

Example 1.8. Let K be a field and let R be the ring of lower triangular 2× 2matrices over K R has two maximal left ideals I1 =

(K 0K 0

)and I2 =

(0 0K K

). The simple

left R-modules S1 = R/I1 and S2 = R/I2 are non isomorphic. Now let M = I1 ⊕ S2 ⊕S1 ⊕ I2.

Notice that R ↪→ M . Hence, � �M� = � �R� = R-Mod. Hence, as S2 is a singularR-module, then S2 is singular in � �M�. Thus, M is not projective in � �M�.

On the other hand, it is clear that S2 is a fully invariant submodule of M . Nowwe consider the module M/S2 I1 ⊕ S1 ⊕ I2. We also note that R ↪→ M/S2. Thus,� �M/S2� = R-Mod. Since I1 is direct summand of R, then I1 is a projective modulein R-Mod. Hence, I1 is a projective module in � �M/S2�.

I2 =(0 0K 0

)⊕(0 00 K

)and

(0 0K 0

)

(0 00 K

) S1

As S1 is a projective simple R-module, then I2 is a projective R- module. Therefore,M/S2 is a projective R-module. Hence, M/S2 is projective in � �M/S2�.

Lemma 1.9. Let M ∈ R-Mod and let L ⊆ N be submodules of M . Let K ∈ R-Modsuch that NMK = 0, then �N/L�M

LK = 0.

Proof. Let f � ML→ K be a morphism and let � � M → M/L denote the

natural projection, then the morphism f � � ∈ Hom �M�K�. Since NMK = 0, then�f � �� �N� = 0. Hence, we have that 0 = f �� �N�� = f �N/L�. Thus, �N/L�M

LK = 0.

�

2. � -FBN MODULES

Definition 2.1. Let M ∈ R-Mod and � ∈ M-tors. The module M is �-boundedif every �-pure essential submodule of M contains a non-zero fully invariantsubmodule of M . M is fully �-bounded if for every P prime in M submodule, themodule M/P is �-bounded.

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Example 2.2. Let R = , and p ∈ R be a prime number. Let M = p� and � =�(p�

) ∈ M-tors. We claim that M is �-bounded. In fact, let 0 �= N � M , it is clearthat N is an essential submodule of M . Moreover, since M/N p� , then N is a�-pure submodule of M . On the other hand, we know that N is a fully invariantsubmodule of M . Thus, M is �-bounded. Also notice that M = p� does not haveprime in M submodules since

(pn

)M

(pm

) = 0 for all n, m ∈ �. Therefore, M isfully �-bounded.

Definition 2.3. A hereditary torsion theory � on � �M� is N -invariant for a fullyinvariant submodule N of M if the module N

NMDis �-torsion for any �-dense

submodule D of M . If � is N -invariant for every fully invariant submodule N of M ,then � is called FIS-invariant.

We observe that � and � are FIS-invariant torsion theories.In the case that M = R, the concept of FIS-invariant torsion theory coincides

with the concept of ideal invariant torsion theory introduced in [11].Notice that if N is a fully invariant submodule of M and if � ∈ M-tors is such

that NMD = N ∩D for all �-dense submodule D of M , then � is N -invariant. Infact, we have that N

NMD= N

N∩D N+DD

∈ ��. Moreover, if NMD = N ∩D for all fullyinvariant submodules N of M and for all �-dense submodules D of M , then � isFIS-invariant.

Remark 2.4. In Wisbauer [19, 37.4] the author shows that if the R-module Mis a submodule of a direct sum of finitely presented modules in � �M�, then Mis regular in � �M� if and only if every finitely generated submodule of M (orM���) is a direct summand. We claim that if M is a submodule of a direct sumof finitely presented modules in � �M� and M is regular in � �M�, then NML =N ∩ L for all fully invariant submodules N of M and for all submodules L ofM . In order to see this, let N be a fully invariant submodule of M and L ⊆ Ma submodule. Then NML ⊆ N ∩ L. Now if x ∈ N ∩ L, then Rx ⊆ N ∩ L. On theother hand, we know that Rx is a direct summand of M , hence there exists asubmodule K of M such that Rx⊕ K = M . Thus we can define the morphismf = i � h � � � M → L, where � � M → M/K is the natural projection, h � M/K →Rx is isomorphism and i � Rx → L is inclusion. Thus f �Rx� = �i � h � �� �Rx� =�i � h� (Rx+K

K

) = i �Rx� = Rx ⊆ L. Therefore x ∈ NML. Hence N ∩ L ⊆ NML. ThusNML = N ∩ L. Moreover notice that if M is a submodule of a direct sum of finitelypresented modules in � �M� and M is regular in � �M�, then � is FIS-invariant for all� ∈ M-tors.

Proposition 2.5. Let � ∈ M-tors, such that � is FIS-invariant. If N ∈ � �M� andN ∈ ��, then AnnM �N ′� = AnnM �N� for all submodules N ′ of N such that N/N ′ ∈ ��.

Proof. Since N ′ ⊆ N , then AnnM �N� ⊆ AnnM �N ′�.Now let K = AnnM �N ′�, then KMN

′ = 0. As � is FIS-invariant. and N ′ is�-dense in N , then K

KMN ′ ∈ ��. Thus K ∈ ��. We claim that KMN = 0. In fact, letf ∈ Hom �M�N�, then f �K� = 0 since N ∈ ��. Therefore KMN = 0. Hence K ⊆AnnM �N�. Thus AnnM �N� = AnnM �N ′�.

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The definition of AssM �N� was given in Castro et al. [9, Definition 4.3]. Weinclude here this definition for the convenience of the reader.

Definition 2.6. Let N ∈ ��M�. A proper fully invariant submodule K of M is saidto be associated to N if there exists a non-zero submodule L of N such thatAnnM�L

′� = K for all non-zero submodules L′ of L.By [9, Proposition 1.16], we have that K is prime in M . We denote by AssM �N�

the set of all prime in M submodules associated to N . Also note that, if N is auniform module, then AssM�N� has at most one element. If N is a uniform moduleand AssM�N� �= ∅, we denote assM�N� the unique prime associated to N .

We denote by ���M� a complete set of representatives of the isomorphismclasses of indecomposable �-torsion free injective modules in ��M�. Notice that if Mhas �M -Gabriel dimension, then ���M� �= ∅.

Let Spec��M� denote the set of �-pure prime in M submodules.Let �� � ���M� −→ Spec��M� defined by ���E� = assM�E�. We will say that M

has local �M -Gabriel correspondence if �� is a bijective function.

Proposition 2.7. Let M ∈ R-Mod and � ∈ M-tors. If M is �-noetherian, thenAssM�N� �= ∅ for all 0 �= N ∈ ��.

Proof. We consider the family � = �AnnM �N ′� �N ′ ⊆ N. By [5, Definition 1.1]we have that AnnM �N ′� = ∩ �ker f � f ∈ Hom �M�N ′�. As N ′ ∈ ��, ker f is �-purein M for all f ∈ Hom �M�N ′�. Thus, AnnM �N ′� is �-pure in M . Inasmuch asM is �-noetherian, the family � has maximal elements. Let L ⊆ N such thatAnnM �L� is maximal in � , then AnnM �L′� = AnnM �L� for all non-zero submoduleL′ of L. Hence, by [9, Proposition 1.16] AnnM �L� is prime in M and AnnM �L�∈ AssM�N�. �

Lemma 2.8. Let M ∈ R-Mod and let L ⊆ N be submodules of M . Let K ∈ R-Modsuch that NMK = 0, then �N/L�M

LK = 0.

Proof. Let f � ML→ K be a morphism and let � � M → M/L denote the natural

projection, then f � � ∈ Hom �M�K�. Since NMK = 0, then �f � �� �N� = 0. Hence,we have that 0 = f �� �N�� = f �N/L�. Thus, �N/L�M

LK = 0.

In the sequel of this article, we will suppose that M is projective in � �M� andHom �M�X� �= 0 for all X ∈ � �M�. Then, notice that M is generator of the category� �M�.

Theorem 2.9. Let M ∈ R-Mod and � ∈ M-tors. Suppose that � is FIS-invariantand M is �-noetherian. If M has local �M -Gabriel correspondence, then M is fully�-bounded.

Proof. Let P be a submodule prime in M . Suppose that there exists a �-pureessential submodule N/P of M/P, such that N/P does not contain non-zero fullyinvariant submodules of M/P. As M is �-noetherian, then M/P is �-noetherian.Hence, we may assume that N/P is maximal with respect to the following

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4612 PÉREZ AND MONTES

property: N/P ⊆ess M/P, N/P is a �-pure submodule of M/P and N/P does notcontain non-zero fully invariant submodules of M/P. We claim that M/P

N/Pis an

uniform module. In fact, suppose that M/P

N/Pis not an uniform module, then there

are submodules 0 �= N ′/PN/P

and 0 �= N ′′/PN/P

of M/P

N/Psuch that N ′/P

N/P∩ N ′′/P

N/P= 0. Hence,

there exist pseudocomplements N ′/PN/P

and N ′′/PN/P

of N ′/PN/P

and N ′′/PN/P

, respectively, such

that N ′/PN/P

∩ N ′′/PN/P

= 0 and N ′/PN/P

⊕ N ′′/PN/P

⊆essM/P

N/P. Moreover, N ′/P

N/Pand N ′′/P

N/Pare �-pure

submodules of M/P

N/P. Thus, N ′/P and N ′′/P are �-pure submodules of M/P.

On the other hand, we know that N/P ⊆ess M/P. Then N ′/P ⊆ess M/P andN ′′/P ⊆ess M/P. Since N/P � N ′/P and 0 �= N/P � N ′′/P, then by maximality ofN/P we have that there exist 0 �= K′/P ⊆ N ′/P and 0 �= K′′/P ⊆ N ′′/P such thatK′/P and K′′/P are fully invariant submodules of M/P. On the other hand, as N ′/P

N/P∩

N ′′/PN/P

= 0, then N ′/P∩N ′′/PN/P

= 0. Hence, N ′/P ∩ N ′′/P = N/P. Now, �K′/P�M/P�K′′/P� ⊆

N ′/P ∩ N ′′/P = N/P ⊆ M/P. By [15, Proposition 18] we have that M/P is a primemodule. Thus, by [15, Definition 16] 0 is prime in M/P. Thus, �K′/P�M/P�K

′′/P� �= 0,a contradiction, since �K′/P�M/P�K

′′/P� is a fully invariant submodule of M/P andN/P does not contain non-zero fully invariant submodules of M/P. Therefore, M/P

N/P

is uniform. Inasmuch as M is �-noetherian, then by Proposition 2.7 AssM(M/P

N/P

) �=∅. We claim that AssM

(M/P

N/P

) = {P}. In fact, as M is a projective module in

�[M], then by [5, Proposition 5.5] PM

(M/P

) = 0. Hence, we have that PM

(M/P

N/P

) =0. Suppose that AssM

(M/P

N/P

) = {P ′}, then there exists 0 �= M ′/P

N/P⊆ M/P

N/Psuch that

AnnM

(M ′′/PN/P

) = P ′ for all M ′′/PN/P

⊆ M ′/PN/P

. As PM

(M/P

N/P

) = 0, then by [9, Proposition

1.3] PM

(M ′/PN/P

) = 0. Thus, P ⊆ P ′. Now, let M ′/P the �-purification of M ′/P in

M/P, then M ′/PN/P

/M ′/PN/P

M ′/PM ′P ∈ ��. Since

M ′/PN/P

⊆ M/P

N/P, then M ′/P

N/P∈ ��. Since � is FIS-

invariant, then by Proposition 2.5 we have that AnnM

(M ′/PN/P

) = AnnM

(M ′/PN/P

). Thus,

AnnM

(M ′/PN/P

) = P ′. On the other hand, we have that N/P ⊆ess M/P, so M ′/P ⊆ess

M/P. Since N/P � M ′/P and M ′/P is a �-pure submodule of M/P, then bymaximality of N/P there exists a 0 �= T/P ⊆ M ′/P such that T/P is a fullyinvariant submodule of M/P. By [9, Proposition 1.3] we have that �P/P ′�M/P�T/P� ⊆�P/P ′�M/P�M

′/P�. Since AnnM

(M ′/PN/P

) = P ′, then P ′M

(M ′/PN/P

) = 0. So by Lemma 2.8

we have that(P′P

)M/P

(M ′/PN/P

) = 0. By Proposition 1.5 M/P is a projective modulein � �M/P�. Thus, by [5, Proposition 5.5] we have that �P ′/P�M/P

(M ′/P

) ⊆ N/P.Therefore, �P ′/P�M/P �T/P� ⊆ N/P. As P ′/P is a fully invariant submodule of M/P,then �P ′/P�M/P �T/P� is a fully invariant submodule of M/P. By [15, Proposition18] we have that M/P is a prime module, then P ′/P = 0, hence, P = P ′. Thus,P = AnnM

(M ′/PN/P

). So AssM

(M/P

N/P

) = �P. Also notice that P is a �-pure submodule of

M since M ′/PN/P

⊆ M/P

N/P∈ ��. It is clear that AssM

( (M/P

N/P

)) = �P. As M is �-noetherian

and M/P ∈ ��, then M/P contains a cocritical submodule C/P. Hence, C/P is anindecomposable �-torsion free injective module in � �M�. By [9, Proposition 4.5] wehave that AssM �M/P� = �P. So by Proposition 2.7, AssM �C/P� �= ∅. Hence, by [9,Proposition 2.4] AssM �C/P� = �P. Thus, AssM�C/P� = �P. By hypothesis M has

local �M -Gabriel correspondence and ��

( (M/P

N/P

)) = P = ���C/P�, then(M/P

N/P

) C/P.On the other hand, we have N/P ⊆ess M/P, then M/P

N/Pis �M/P�-singular. Now, as

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FBN MODULES 4613

M/P is a �-noetherian and prime module, then by Proposition 1.3 we have thatM/P is a non M/P-singular module. Hence, C/P is a non M/P-singular module.Since (M/P

N/P

) C/P then there exists a non zero submodule C ′/P of M/P such thatC ′/P ↪→ M/P

N/P, a contradiction. Therefore, M is fully �-bounded. �

Theorem 2.10. Let M ∈ R-Mod and � ∈ M-tors. Suppose that M is �-noetherian andprojective in ��M�. If M is fully �-bounded, then M has �M -Gabriel correspondence.

Proof. As M is �-noetherian, then we have that M has �M -Gabriel dimension, thenby [9, Theorem 4.9] it suffices to show that M has �-�M -dimension. Let �′ ∈ M-torssuch that � ≤ �′ < �. As M has �M -Gabriel dimension, there exists C ∈ � �M� suchthat C is a �′-cocritical module. So C ∈ ��. Inasmuch as M is �-noetherian, thenAssM �C� �= ∅. Let P be a prime in M such that AssM �C� = �P � then there exists0 �= C ′ ⊆ C such that AnnM �C ′′� = P for all 0 �= C ′′ ⊆ C ′. Thus, PMC

′ = 0. Since Mis generator of � �M�, then there is a non-zero morphism f � M → C ′. If K = ker f ,then M/K ↪→ C ′. Thus, PM �M/K� = 0. By [5, Proposition 5.5] we have that PMM ⊆K. Since P is prime in M , then P is a fully invariant submodule of M . So PMM = P.Hence, P ⊆ K. Thus, K/P ⊆ M/P. We claim that K/P is not an essential submoduleof M/P. In fact, suppose that K/P ⊆ess M/P. As M is fully �-bounded, then thereexists 0 �= L/P ⊆ K/P such that L/P is a fully invariant submodule of M/P. Hence,L is a fully invariant submodule of M . Since M/K ↪→ C ′, then AnnM �M/K� = P.As LMM = L ⊆ K and M is projective in � �M�, then by [5, Proposition 5.5] wehave that LM �M/K� = 0. Thus, L ⊆ AnnM �M/K� = P. So L = P, a contradiction.As K/P is not essential in M/P, then there exists a non zero submodule K′/P ofM/P such that K′/P is a pseudocomplement of K/P. Hence, K/P ⊕ K′/P ↪→ M/P.Thus, K′/P ↪→ M/K. Therefore, K′/P ↪→ C ′. Since C ′ is �′-cocritical module, then� �K′/P� = � �M/K� = � �C ′�. On the other hand, by [9, Proposition 2.7], we havethat M/P is a � �M/P�-�M -module. Since K′/P ⊆ M/P, then by [9, Proposition 2.2],� �K′/P� = � �M/P�. Therefore, � �M/P� = � �C ′� = � �C�. Since C is a �′-cocriticalmodule, then by [9, Definition 2.1] C is a �′-�M -module. Thus, by [9, Definition 2.8],we have that C is a �′-�M -module. Therefore, by [9, Corollary 3.6] M has �-�M -dimension.

Corollary 2.11. Let M ∈ R-Mod and � ∈ M-tors. Suppose that � is FIS-invariant andM is �-noetherian. Then the following conditions are equivalent:

(i) M is fully �-bounded.(ii) M has �-�M -dimension.

Proof. �i� ⇒ �ii� It follows from the proof of Theorem 2.10.

�ii� ⇒ �i� As M is �-noetherian, then we have that M has �M -Gabrieldimension, then by [9, Theorem 4.9], we have that M has �M -Gabrielcorrespondence. So, by Theorem 2.8 we have that M is fully �-bounded. �

Notice that if � ∈ M-tors is FIS-invariant and if M is �-noetherian, then eachone of the conditions in [9, Theorem 4.9] are equivalent to the condition of M beingfully �-bounded.

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4614 PÉREZ AND MONTES

Theorem 2.10 is not true in general. Consider the Example 2.2. In this example,� = �

(p�

)and M = p� is fully �

(p�

)-bounded. Moreover, p� does not have

prime submodules, but p = p� is an indecomposable �-torsion free injectivemodule in ��p� �.

The following example shows that the conclusion of Theorem 2.10 need notbe true without the hypothesis that M is projective in ��M�, not even when M isassumed to be noetherian.

Example 2.12. Let R = 2 � �2 ⊕ 2�, the trivial extension of 2 by 2 ⊕ 2.This ring can be described as R = {(

a �x�y�0 a

) � a ∈ 2� �x� y� ∈ 2 ⊕ 2

}. R has only

one maximal ideal I = {(0 �x�y�0 0

) � �x� y� ∈ 2 ⊕ 2

}and it has three simple ideals: J1,

J2, J3, which are isomorphic, where J1 ={(

0 �0�0�0 0

)�(0 �1�0�0 0

)}, J2 =

{(0 �0�0�0 0

)�(0 �0�1�0 0

)},

and J3 ={(

0 �0�0�0 0

)�(0 �1�1�0 0

)}. Then the lattice of ideals of R has the following form:

R is artinian and R-Mod has only one simple module. Let S be the simplemodule. By [16, Theorem 2.13], we know that there is a lattice anti-isomorphismbetween the lattice of ideals or R and the lattice of fully invariant submodules ofE �S�. Thus, the lattice of fully invariant submodules of E �S� has tree maximalelements K, L, and N . Moreover, E �S� contains only one simple module S.Therefore, the lattice of fully invariant submodules of E �S� has the following form:

Let M = E �S�. As K ∩ L = S and KML ⊆ K ∩ L, then KML ⊆ S. On the other

hand, we consider the morphism, f � M�→ �M/N� S

i↪→ L, where � is the natural

projection and i is the inclusion. Thus, f �K� = S. Therefore, S ⊆ KML. Thus, wehave that KML = S. Thus, KML ⊆ N but K � N and L � N . Therefore, N is notprime in M . Analogously, we prove that neither K nor L are prime in M . Also, wenote that KMK = S and SMK = 0. Thus, AnnM

�K� = S.

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FBN MODULES 4615

We claim that KMS = S. In fact, if � is the natural projection from M ontoM/N , then � �K� = S since �M/N� S. Thus, KMS = S. Analogously, we prove thatLMS = S and NMS = S. Moreover, it is not difficult to prove that AnnM

�S� = K ∩L ∩ N = S and that S is not a prime in M submodule. Soc �R� = J1 + J2 + J3 =J1 ⊕ J2 and Soc �R� ⊂ess R. As S is the only simple module, then Soc �R� S ⊕S. So E �S ⊕ S� = E �R�. Thus, E �R� = E �S�⊕ E �S� = M ⊕M . Therefore, � �M� =� �R� = R-Mod. Moreover, Hom �M�K� �= 0 for all K ∈ � �M� but M is not agenerator of � �M�.

Now let 0 �= M ′ a submodule of M . As S ⊆ess M , then S ⊆ M ′. Therefore, everysubmodule of M contains a fully invariant submodule of M .

From the previous arguments, we can conclude that:

(1) M does not have prime in M submodules.(2) M is the only one indecomposable injective module in ��M�.(3) M is noetherian and M is not projective in � �M�.(4) M is fully �-bounded.(5) M does not have �M -Gabriel correspondence.

Remark 2.13. If M is a noetherian module such that each submodule of M isfully invariant, then M has �M -Gabriel correspondence for all � ∈ M-tors. In fact,let � ∈ M-tors, since each submodule of M is fully invariant, then M is fully �-bounded. Now, by Theorem 2.10, we have that M has �M -Gabriel correspondence.For instance, if M is a left R-module such that M = S1 ⊕ S2 ⊕ · · · ⊕ Sn whereS1� S2� � Sn are non-isomorphic simple left R-modules, then M is noetherian andeach submodule of M is fully invariant.

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