On Q-Injective, Duo Submodules of C 1 -Module

This note investigates modules having quasi-injective and duo submodules. We introduce a new generalization of 𝐶 ! -module. The main method that was adopted in this generalization is how to obtain a submodule 𝒩 in ℳ having the characteristic Quasi-injective. We investigate the relationship between pseudo-injective module and Quasi-injective property of 𝐶 ! -module. Finally, we introduce a new relationship between Quasi-injective submodule and anti-hopfian module.


INTRODUCTION
All the modules in this paper have a unity.Many searchers studied Quasi-injective and injective modules in details.Here we study Quasi-injective of any submodule  of ℳ.In [1], An R-module P is a projective module if there exists an Rmodule Q such that P ⊕ Q is a free R-module; also more details about injective and projective module can find it in same reference.In [2], we can find the definition of a Quasiinjective module (briefly Q-injective).Also, in [3], the author said ℳ is pseudo-injective module (p-injective module) if ∀  ≤ ℳ, each R-isomorphism :  → ℳ can be extended to an R-endomorphism of ℳ.In [4], A module ℳ is called uniform if  ! and  " are non-zero submodules of ℳ;  !∩  " ≠ 0 the intersection of any two non-zero submodules is nonzero, equivalently, ℳ is uniform if 0 ≠  ≤ #$$ ℳ.In [5],  ≤ ℳ is called stable if for each R-homomorphism :  → ℳ implies () ⊆ , and an R-module ℳ is called fully stable in case every submodule of ℳ is stable.*Department of Mathematics / College of Education for Pure Sciences / University of Anbar, Iraq, Tel: +96407804621498 E-mail address: abd19u2007@uoanbar.edu.iq In this article, we investigate some facts about any submodule  of  !-module ℳ like Q-injective and duo properties.Also we use other properties in order to satisfy the same goal such as hopfian, anti-hopfian and self-injective modules.

2.PSEUDO-INJECTIVE and QUASI-INJECTIVE SUBMODULES
In this section, we will study two important properties of submodule  of ℳ namely Quasi-injective and P-injective.Via this submodule, we obtain a new characterization of  !module.Moreover; we should provide another property namely fully invariant of this submodule.Note that Qinjective itself injective.Definition 2.1.[1].An R-module ℳ is called injective if for every monomorphism ℎ: ℳ !→ ℳ " and homomorphism : ℳ !→ ℳ % there exists a homomorphism : ℳ " → ℳ % such that  ∘ ℎ = .Definition 2.2.[2].Let ℳ be an R-module.Then ℳ is said to be Q-injective if for each submodule  of ℳ and Rhomomorphism :  → ℳ can be extended to an Rendomorphism of ℳ. Definition 2.3.[3].An R-module ℳ is called pseudoinjective, if for every submodule  of ℳ, each R-isomorphism :  → ℳ can be extended to an Rendomorphism of ℳ. Lemma 2.4.[6].Let ℳ be an R-module over P.I.D.If ℳ is pseudo-injective module, so it is a Q-injective.Now we need to find a submodule  of ℳ such that  is a Q-injective with invariant property.From [3], any pseudoinjective module over P.I.D is a Q-injective; this means if ℳ is a module on P.I.D, so  ≤ ℳ on P.I.D, but ℳ is pseudoinjective;  is a pseudo-injective and hence  is a Qinjective.Note that to understanding lemma (2.4), we can see [7].
The following theorem explain the relationship between pseudo-injective and  !-module over P.I.D. Theorem 2.5.Let a ring R be a P.I.D.If ℳ is a pseudoinjective  !-module over R, then any submodule  ≤ ℳ is a Q-injective and f() ⊆ ; so ℳ is Q-injective-duo- !module.Proof: Suppose that a module ℳ is pseudo-injective.Let us take  ≤ ℳ.We have ℳ any module on P.I.D So also  ≤ ℳ on P.I.D.But ℳ is pseudo-injective, then  is pseudoinjective over P.I.D. Hence  is Q-injective with f() ⊆  imply  is fully invariant (duo) submodule of ℳ.Thus ℳ is Q-injective-duo- !-module.Now we introduce another way to obtain any submodule  of  !-module ℳ and be Q-injective.This way depends on new domain namely Dedekind domain.(R is a Dedekind domain if it is integrally closed, Noetherian and if 0 ≠ p is a maximal; p is prime ideal).So if R is a Dedekind domain, then it is a UFD if and only if R is P.I.D. See the next Lemma: Therefore  is a duo submodule of ℳ. Lemma 2.8.[7].Let ℳ be an R-module.If the following statements are true: (1)-R is Multiplication ring; (2)-ℳ is P-injective; (3)-(ℳ) = ℳ; then ℳ is Q-injective and so  is also Q-injective.Theorem 2.9.Let ℳ be a module over a ring R. If: (1)-R is multiplication ring; (2)-(ℳ) = ℳ; (3)-ℳ is stable; (4)-ℳ is  !-module and Pseudo-injective; then ℳ is Q-injective-duo- !-module.Proof: Assume that (ℳ) = ℳ and R is a multiplication ring.Then from [8], () =  (any submodule of torsion module is torsion).Since ℳ is P-injective, then ℳ is a Qinjective and hence  is P-injective and Let :  → ℳ be an R-homomorphism.So () = 0 or () = .Suppose that () = , so g can be extended to homomorphism ℎ: ℳ → ℳ.Now if () = 0, so g is one to one and can be extended to R-homomorphism from ℳ → ℳ (ℳ is Pseudo-injective).Hence  is Q-injective.Corollary 2.12.Let ℳ be a  !-pseudo-injective R-module.If () ⊆ ,  ≤ #$$ ℳ and (ℳ) = 0; then ℳ is Qinjective-duo- !-module.Now we present another way in order to obtain that any submodule  ≤ ℳ is a Q-injective.But before that we need to present some important definitions that are closely related to the mentioned way.Firstly, a concept of Stable-Q-injective was explained in [6].Let :  →  ∋ () ⊆ .Then ℳ is called stable module.So if every  ≤ ℳ is stable this means ℳ is fully stable module (F-stable).If  ≤ ℳ is stable and can be extended R-homomorphism Note that: 1.

2.
If (ℳ) = 0, then a module ℳ is called torsion-free-module. Lemma 2.13.[6].Let ℳ be a stable-Q-injective R-module.If ℳ is an injective R-module, then it is Q-injective.Theorem 2.14.Let ℳ be a  !-module.If ℳ is a F-stable and stable-Q-injective; then ℳ is Q-injective-duo-C1-module. Proof: Let  ≤ ℳ and let :  → ℳ be an R- Recall that a ring R is called Quasi-Frobenius (QF-ring) if every projective module is injective; or every injective module is discrete.From [11], every projective-module is injective and then every injective-module is Q-injective.Corollary 2.17.Let ℳ be a  !-module over QF-ring.If ℳ is a projective module and stable in R, then ℳ is Q-injectiveduo- !-module ( is Q-injective submodule).Proof: Let R be a QF-ring.Since ℳ is a projective Rmodule, then ℳ is an injective module and hence any submodule  of ℳ is Q-injective.Note that ℳ is stable module; so for :  → ℳ be a homo.we get () ⊆ .Thus ℳ is Q-injective-duo- !-module.
19.For a ring R, we have  * is a semi-simple if and only if R is a semisimple and so any module ℳ over R is a semisimple module.Proof: We need to prove the following, 1.  * semisimple if and only if R is semisimple.

2.
ℳ is a semisimple module over R. From [11], we can get the proof of (1).Now we need to proof (2): If  * is a semisimple and if ℳ = ℳ * ∋  ∈ ℳ, then R is a semisimple as an epimorphic image of  * .So ℳ = ∑ ,  ∈ ℳ as a sum of semisimple module is again semisimple.Lemma 2.20.Let a ring R be a semisimple, and ℳ be an Rmodule.Then every submodule  ≤ ℳ is Q-injective.
Proof: Since R is a semisimple ring, then every module ℳ over R is a semisimple.So  ≤ ℳ is a direct summand.Hence ℳ is injective R-module.But every injective R-module is a Q-injective.Thus  is Q-injective.Theorem 2.21.Let R be a semisimple ring and ℳ is an Rmodule.If ℳ is  !-module and stable; then it is Q-injectiveduo- !-module.Proof: It is clear that from lemma (2.20),  ≤ ℳ is Qinjective.But ℳ is a stable, then ∃ :  → ℳ ∋ () ⊆ .So  is a fully invariant and hence ℳ is a duo ( is a duo submodule).We have ℳ is  !-module.So it is  !-module.Thus  is Q-injective of ℳ. Corollary 2.22.Let ℳ be an R-module.If: (1)-ℳ is projective module; (2)-ℳ is a simple module; (3)-ℳ is Q-injective; then  is Q-injective and duo submodule of  !-module.Proof: It is clear that projective module means  !-module.Also, if a module ℳ is simple, then ℳ is duo-module.( ≤ ℳ ∋  is fully invariant; f() ⊆  and :  → ℳ is an Rhomomorphism).Now from condition (3), we have ℳ is Quasi-projective.So ℳ is a Q-injective and hence  is a Qinjective of  !-module.
Recall that any ring R is called V-ring if every simple Rmodule is injective [12].Corollary 2.23.Let ℳ be a  !-R-module over V-ring.Then ℳ is Q-injective-duo- !-module.

2.HOPFIAN, SELF-INJECTIVE MODULES AND Q-INJECTIVE SUBMODULE
From [13], a module ℳ is called self-p-injective if ℳ satisfy the following condition; every homomorphism from a projection invariant submodule of ℳ to ℳ can be lifted to ℳ.
Therefore, every simple module is indecomposable, but the converse is not true.Theorem 3.3.Let ℳ be an indecomposable self-P-injective R-module.Then any  !-module is Q-injective-duo- !module.
Proof: From definition of self-p-injective, there exists K submodule of ℳ such that K is fully invariant.Assume that ℳ is indecomposable module, so every submodule of ℳ is projective invariant.Then ℳ is Q-injective.Thus ℳ is Qinjective-duo- !-module.
Recall that any module ℳ is called Hopfian if every surjective f in (ℳ) is isomorphism and a non simple module is called anti-Hopfian if proper submodule of ℳ is a non-Hopfian kernel such that a submodule  of ℳ is non-Hopfian kernel (for ℳ) if there exists an isomorohism ℳ/  to ℳ [14].Or an R-module ℳ is anti-Hopfian if ℳ is non simple and all nonzero factor modules of ℳ are isomorphic to ℳ; that is for all  ≤ ℳ, ℳ/ ≅ ℳ Proof: From [14], ℳ is anti-Hopfian module.Since ℳ1 and ℳ2 are simple modules, then ℳ is a simple module and so it is a duo module ( is a duo submodule).From lemma (3.5), the proof is completed.Corollary 3.7.Let R be a Dedekind domain, and ℳ is C1module with (ℳ) ≠ ℳ.  ℳ ≅ / " ∋  is a non-zero ideal of R and  is duo submodule of ℳ, then  ≤ ℳ is Qinjective in C1-module.Proof: From [14] and Lemma (3.5).

CONCLUSIONS
This paper investigated modules having a submodule are duo and Quasi-injective properties.Tow generalization of  !module have been studied.We proved that any module has pseudo-injective,  is essential in ℳ and stable, this mean ℳ is a Quasi-injective-duo- !-module where R is a Dedekind domain.Also same goal can obtained it if ℳ is a projective and stable with  is an essential in ℳ.
module.From definition of fully invariant submodule and definition of stable, we find the two meanings are same.