GF-Regular Modules

We introduced and studied GF-regular modules as a generalization of π-regular rings to modules as well as regular modules (in the sense of Fieldhouse). An R-moduleM is called GF-regular if for each x ∈ M and r ∈ R, there exist t ∈ R and a positive integer n such that rntrnx = rx. The notion of G-pure submodules was introduced to generalize pure submodules and proved that an RmoduleM isGF-regular if and only if every submodule ofM isG-pure iffMM is aGF-regular RM-module for each maximal ideal M of R. Many characterizations and properties of GF-regular modules were given. An R-moduleM isGF-regular iff R/ann(x) is a π-regular ring for each 0 ̸ = x ∈ M iff R/ann(M) is a π-regular ring for finitely generated moduleM. IfM is a GF-regular module, then J(M) = 0.


Introduction
Throughout this paper, unless otherwise stated,  is a commutative ring with nonzero identity and all modules are left unitary.For an -module , the annihilator of  ∈  in  is ann  () = { ∈  :  = 0}.The symbol ◻ stands for the end of the proof if the proof is given or the end of the statement when the proof is not given.
Recall that a ring  is said to be regular (in the sense of von Neumann) if for each  ∈ , there exists  ∈  such that  =  [1].The concept of regular rings was extended firstly to -regular rings by McCoy [2], recall that a ring  is -regular if for each  ∈ , there exist  ∈  and a positive integer  such that     =   [2] and secondly to modules in several nonequivalent ways considered by Fieldhouse [3], Ware [4], Zelmanowitz [5], and Ramamurthi and Rangaswamy [6].In [7], Jayaraman and Vanaja have studied generalizations of regular modules (in the sense of Zelmanowitz) by Ramamurthi [8] and Mabuchi [9].Following [10], we denoted Fieldhouse' regular modules by -regular.An -module  is called -regular if each submodule of  is pure [3].
Dissimilar to the generalizations that have been studied in [7,9] and [8], in this paper a new generalization of -regular rings to modules and -regular modules was introduced, called -reular (generalized -regular) modules.An module  is called -regular if for each  ∈  and  ∈ , there exist  ∈  and a positive integer  such that      =   .A ring  is called -regular if  is -regular as an -module.On the other hand, -regular modules are also a generalization of -regular rings.Thus,  is a -regular ring if and only if  is a -regular -module.Furthermore, we introduced a new class of submodules, named, -pure submodules as a generalization of pure submodules.A submodule  of an -module  is said to be -pure if for each  ∈ , there exists a positive integer  such that  ∩    =   .Recall that a submodule  of an -module  is pure if  ∩  =  for each ideal  of  [11].We find that the relationship between -regular modules and -pure submodules is an analogous relationship between regular modules and pure submodules.
In Section 3.1 of this paper, after the concept of regular modules was introduced, we obtained several characteristic properties of -regular modules.For instance, it was proved that the following are equivalent for an -module : (1)  is -regular; (2) every submodule of  is -pure; (3) /ann() is a -regular ring for each 0 ̸ =  ∈ ; (4) and for each  ∈  and  ∈ , there exist  ∈  and a positive integer  such that  +1  =   .It is also shown that if

The Notion of 𝐺𝐹-Regular Modules and General Results
We start by recalling that an -module  is -regular if each submodule of  is pure [3], and a ring  is -regular if for each  ∈ , there exist  ∈  and a positive integer  such that Proof.Suppose that  is a -regular -module, so for each  ∈  and  ∈ , there exist  ∈  and a positive integer  such that      =   ; hence, (    −   ) ∈ ann() which means that     =   ; therefore, /ann() is a -regular ring.Conversely, suppose that /ann() is a -regular ring for each 0 ̸ =  ∈ , thus for each  ∈ /ann(), there exist  ∈ /ann() and a positive integer  such that     =   ; hence,     −  ∈ ann() which implies that (    −  ) = 0; therefore,  is a -regular -module.
It is clear that every -regular module is -regular, but the converse may not be true in general; for example, by applying Proposition 2 to the -module  4 , we can easily see that it is -regular; however,  4 is not an -regular -module.In fact, the -module   is -regular for each positive integer  [12], while it is not -regular for some positive integer .On the other hand, the -module  is not -regular because for each 0 ̸ =  ∈  we have that ann  () = 0, but /ann  ≃  which is not a -regular ring [12].
(3) A ring  is -regular if and only if  is -regular as an -module.
(4) Every submodule of a -regular module is regular module.In particular, every ideal of a regular ring  is -regular -module.Furthermore, it follows from (1) that if  is an ideal of a -regular ring , then the -module / is -regular.
We have seen previously that every -regular -module is -regular.In the following we consider some conditions such that the converse is true.

Remark 5.
(1) Let  be a reduced ring.An -module  is -regular if and only if  is a -regular -module.
(2) An -module  is -regular if and only if  is a -regular -module and (/ann()) = 0 for each 0 ̸ =  ∈ , where (/ann()) is the prime radical of the ring /ann().Now, we describe -regular modules over the ring of integers .

Proposition 6. A 𝑍-module 𝑀 is 𝐺𝐹-regular if and only if 𝑀 is a torsion 𝑍-module.
Proof.If  is a -regular -module, then by Remark 3(6)  is a torsion -module.Conversely, if  is a torsion module, then ann  () =  for some positive integer ; hence, /ann  () ≃   is a -regular ring for each positive integer  [12], which implies that  is a -regular module.

Proposition 7. Every homomorphic image of a 𝐺𝐹-regular 𝑅module is 𝐺𝐹-regular.
Proof.Let ,   be two -modules such that  is regular and let  :  →   be an -epimorphism.For every  ∈   , there exists  ∈  such that () = .It is clear that ann() ⊆ ann().Define  : /ann() → /ann() by (+ann()) = +ann() for each  ∈ .It is an easy matter to check that  is well defined -epimorphism.Since /ann() is a -regular ring, then /ann() is also a -regular ring [12].Therefore,   is a -regular -module.Corollary 8.The following statements are equivalent for an module : (1) / is a -regular -module for every nonzero submodule  of .
Another characterization of a -regular -module is given in the next result.Proposition 9.An -module  is -regular if and only if for each  ∈  and  ∈ , there exist  ∈  and a positive integer  such that  +1  =   .
Proof.Suppose that  is a -regular -module, so for each  ∈  and  ∈ , there exist  ∈  and a positive integer  such that      =   , then we can take  =  −1 ∈  and hence  +1  =   .Conversely, for each  ∈  and  ∈ , there exist  ∈  and a positive integer  such that

𝐺𝐹-Regular Modules and Purity.
Recall that a submodule  of an -module  is pure in  if each finite system of equations which is solvable in , is solvable in  [13].It is not difficult to prove that  is pure in  if and only if for each ideal  of , ∩ =  [11].This motivates us to introduce the following definition as a generalization of pure submodules.
Definition 10.A submodule  of an -module  is called pure if for each  ∈ , there exists a positive integer  such that  ∩    =   .
It is clear that every pure module is -pure.
The following theorem gives another characterization of -regular modules in terms of -pure submodules.
Theorem 11.An -module  is -regular if and only if every submodule of  is -pure.
Proof.Suppose that  is a -regular -module and let  be any submodule of .For each  ∈  and for some positive integer , let  ∈  ∩   , then there exists  ∈  such that  =   .Since  is -regular, then there exists  ∈  such that    =     .Put  =   , then    =    which implies that  = , but  ∈ , so  =  ∈    and hence  ∩    ⊆   .On the other hand, it is clear that    ⊆  ∩   , thus  ⋂    =    which means that  is a -pure submodule.
Conversely, assume that every submodule is -pure and let  ∈  and  ∈  such that    =  which is a pure submodule of  for some positive integer , then  ∩    =    for each  ∈ .In particular, if  =  we get    ∈  ∩    ⊆    =      which implies that there exists  ∈  such that      =   , so  is a -regular -module.
Corollary 12.An -module  is -regular if and only if for each  ∈ , there exist  ∈  and a positive integer  such that    is a -pure submodule.
Remark 13.Fieldhouse in [11] proved that for a submodule  of an -module , if / is a flat -module, then  is pure.On the other hand, if  is flat and  is pure, then / is flat.So, immediately we have that for a flat -module, if / is a flat -module for each submodule  of , then  is regular -module.It is not difficult to prove that in case of regular modules the converse of the latest statement is true; however, we do not know whether it is true for -regular modules or not.Remark 14.In [14], Mao proved that a right -module  is -flat if and only if there exists an exact sequence 0 →  →  →  → 0 with  free such that for any  ∈ , there exists a positive integer  satisfying  ∩   =   , where (1) a right -module  is said to be generalized -flat (-flat for short) if for any  ∈ , there exists a positive integer  (depending on ) such that the sequence 0 →  ⊗   →  ⊗  is exact [15], (2) a right -module  is -flat [16] or torsion-free [15] if for any  ∈ , the sequence 0 →  ⊗  →  ⊗  is exact.Obviously, every flat module is -flat [16] and every -flat module is -flat [14].
According to the above remark we get the following.
Corollary 15.An -module  is -flat if and only if there exists an exact sequence 0 →  →  →  → 0 with  is a submodule of a free -module  such that  is a -pure submodule.
Corollary 16.For every submodule  of a free -module , if there exists an exact sequence 0 →  →  →  → 0 such that  is a -pure submodule in , then  is a -flat -module if and only if  is -regular.Now, we recall that (1) an -module  is -injective if for every principal ideal  of , every -homomorphism of  into  extends to one of  into  [17].A ring  is called injective if  is -injective as an -module.(2) An -module  is called -injective if for any 0 ̸ =  ∈ , there exists a positive integer  such that   ̸ = 0 and any -homomorphism of   into  extends to one of  into .A ring  is called -injective if  is -injective as an -module [18].injective modules are called -injective modules by some other authors [19][20][21][22].(3) An -module  is called injective (weak -injective) if for any  ∈ , there exists a positive integer  such that every -homomorphism of   into  extends to one of  into  (  may be zero).A ring  is called -injective if  is -injective as an -module [23][24][25].(4) A ring  is called ..if every principal ideal of  is projective.And  is called -ring if for any  ∈ , there exists a positive integer  (depending on ) such that   is projective [26,27].
Note that -injectivity implies -injectivity (or injectivity) and -injectivity, as well as the concept of ..rings implies the concept of -rings.However, the notion of -injective (or -injective) modules is not the same notion of -injective modules.
It is known that a ring  is -regular if and only if every -module is -injective [12,22], so from all the above we conclude the following theorem.
Theorem 17.The following statements are equivalent for a ring .
We end this section by the following two related results.

Proposition 18.
Let  be an -module.If /() is a regular ring, then  is a -regular -module.
In case of finitely generated modules, the converse of Proposition 18 is true.

Proposition 19.
Let  be an -module.If  is a finitely generated -regular -module, then /() is a -regular ring.

𝐺𝐹-Regular Modules and Localization.
In this section we study the localization property and semisimple modules with -regular modules and we give some characterizations of -regular modules in the sense of them.
Theorem 20.Let  be an -module. is a -regular module if and only if  M is a -regular  M -module for each maximal ideal M in .
Proof.Let  be a -regular -module, and let M be any maximal ideal in .Let / ∈  M and / 1 ∈  M , where  ∈ ,  ∈  and ,  1 ∈ −M.So there exist  ∈  and a positive integer  such that      =   .Hence, Conversely, suppose that  M is a -regular  Mmodule.Let  be a submodule of  and let M be a maximal ideal of .By Theorem 11,  M is a -pure submodule of  M ; therefore,  M ∩ (  ) M  M = (  ) M  M for each  ∈  and for some positive integer .But by [28], we have that again by [28], we get that    ∩  =   , which implies that  is a -pure submodule of  and by Theorem 11 , is a -regular -module.
Recall that an -module  is simple if 0 and  are the only submodules of , and an -module  is said to be semisimple if  is a sum of simple modules (may be infinite).A ring  is semisimple if it is semisimple as an -module [29].It is known that over any ring , a semisimple module is regular [4,30], consequently it is -regular.Furthermore, it is known that over a local ring, every -regular module is semisimple [31].We can generalize the latest statement as the following.
Proposition 21.Every -regular module over local ring is semisimple.
Proof.Let M be the only maximal ideal of .Since  is regular, then for each 0 ̸ =  ∈  we have that /ann() is -regular local ring which implies that /ann() is a field [12]; hence, ann() is a maximal ideal, so M = ann() for each 0 ̸ =  ∈ .Therefore, M = ann() = ann().On the other hand, /M ≃ /ann() is a field, which implies that  is a vector space over the field /ann() which is a simple ring.Then  is a semisimple module over the ring /ann().Thus,  is a semismple -module [29].
As an immediate result from Theorem 20 and Proposition 21, we get the following.

Corollary 22.
Let  be an -module. is -regular if and only if  M is a semisimple  M -module for each maximal ideal M of .
We mentioned before that every -regular -module is -regular; the following gives us another condition such that the converse is true.Proof.Assume that  and  are -regular -modules, then for each maximal ideal M in , each of  M and  M is a semisimple module (Proposition 21); hence, it is an easy matter to check that  M +  M is a semisimple module, so  M =  M ⨁  M is a -regular module.Thus,  is a regular module (Theorem 20).The other direction is obtained directly from Proposition 7.
Finally we can summarize that the conditions under which -regular modules coincide with -regular modules and the characterizations of -regular modules, of Section 2 with those of this section, in the following Proposition 25 and Theorem 26, respectively: Proposition 25.An -module  is -regular if and only if  is an -regular module, if any of the following conditions are satisfied.
(1)  is a local ring.
Theorem 26.The following statements are equivalent for a ring .
(5) For each  ∈ , there exist  ∈  and a positive integer  such that    is a -pure submodule.
(6)  is a -flat -module, if for every submodule  of a free -module  there exists an exact sequence 0 →  →  →  → 0 such that  is a -pure submodule in .
(7) If  is a finitely generated -module, then /() is a -regular ring.
(8)  M is a -regular  M -module for each maximal ideal M in .
(9)  M is a semisimple  M -module for each maximal ideal M of .

The Jacobson Radical of 𝐺𝐹-Regular
Modules.Let  be an -module.A submodule  of  is said to be small in  if for each submodule  of  such that  +  = , we have  =  [32].The Jacobson radical of a ring  will be denoted by ().The following submodules of  are equal: (1) the intersection of all maximal submodules of , (2) the sum of all the small submodules of , and (3) the sum of all cyclic small submodules of .This submodule is called the Jacobson radical of  and will be denoted by () [29,32].It is appropriate now to note that for each element  ∈  it may happen that   = 0.But some cases demand that   must be nonzero element.For this purpose we introduce the following concept.Definition 27.An -module  is called -regular if for each 0 ̸ =  ∈  and  ∈ , there exist  ∈  and a positive integer  with   ̸ = 0 such that      =   .A ring  is called -regular if it is -regular as an -module.
It is clear that -regularity implies -regularity and they are coincide if  is a reduced ring.Proposition 28.Let  be an -regular -module, then ().= 0.
Recall that an -module  is faithful if for every  ∈  such that  = 0 implies  = 0 [29], or equivalently, an module  is called faithful if ann() = 0 [33].It is suitable to mention that, in general, not every module contains a maximal submodule; for example,  as -module has no maximal submodule.So we have the next two results, but first we need Lemma 33 which is proved in [29].
Lemma 33.An -module  is semisimple if and only if each submodule of  is direct summand.Proposition 34.Let  be a -regular -module, then () = 0.
Proof.Since  is a -regular -module, then  M is a semisimple  M -module for each maximal ideal M of  (Corollary 22).Since each cyclic submodule of  M is direct summand (Lemma 33), then it cannot be small; therefore, the Jacobson radical of a semisimple module is zero, so ( M ) = 0 for each maximal ideal M of .On the other hand, () M ⊆ ( M ) [28], thus () M = 0 for each maximal ideal M of , and hence () = 0 [28].
Corollary 36.Let  be a -regular -module, then for each 0 ̸ =  ∈ , there exist a maximal submodule M such that  ∉ M.
Corollary 37. Let  be a -regular -module, then every proper submodule of  contained in a maximal submodule.
Proof.Let  be a proper submodule of .Since  is a -regular -module, then / ̸ = 0 is -regular (Proposition 7), so / contains a maximal submodule (Corollary 35), which means that there exists a submodule  of  such that  ⊆ , / is a maximal submodule of /; therefore,  is a maximal submodule of  and contains .
Corollary 38.Every simple submodule of a -regular module is direct summand.
Proof.Let  be a simple submodule of a -regular module , then  is cyclic; say  = , then there exists a maximal submodule M of  such that  ∉ M (Corollary 37).It is clear that  = M + .Now, if  ∩ M ̸ = (0), then ∩M =  because  is a simple submodule.Thus,  ∈ M which is a contradiction, so  =  ⨁ M.

Corollary 29 .Corollary 32 .
If  is a faithful -regular -module, then () = 0.Corollary 30.Let  be a reduced ring and  be a -regular -module, then () ⋅  = 0.Corollary 31.Let  be any ring such that () is a reduced ideal of  and let  be a -regular -module, then ()⋅ = 0. Let  be a reduced ring.If  is a faithful regular -module, then () = 0.

Corollary 23 .
Let  be a local ring.An -module  is regular if and only if  is a -regular -module.An -module  =  ⨁  is -regular if and only if  and  are -regular -modules.