# Existence of Covers and Envelopes of a Left Orthogonal Class and Its Right Orthogonal Class of Modules

Authors: Arunachalam Umamaheswaran; Ramalingam Udhayakumar; Chelliah Selvaraj; Kandhasamy Tamilvanan; Masho Jima Kabeto
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, we investigate the notions ofX⊥-projective, X-injective, and X-flat modules and give some characterizations of these modules, where X is a class of left modules. We prove that the class of all X⊥-projective modules is Kaplansky. Further, if the class of all X-injective R-modules is contained in the class of all pure projective modules, we show the existence of X⊥-projective covers and X-injective envelopes over a X⊥-hereditary ring. Further, we show that a ring R is Noetherian if and only if W-injective R-modules coincide with the injective R-modules. Finally, we prove that if W⊆S, every module has a W-injective precover over a coherent ring, where W is the class of all pure projective R-modules and S is the class of all fp−Ω1-modules.


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## Body

## 1. Introduction

Throughout this paper,R denotes an associative ring with identity and all R-modules, if not specified otherwise, are left R-modules. R-Mod denotes the category of left R-modules.The notion offp-injective modules over arbitrary rings was first introduced by Stenström in [1]. An R-module M is called fp-injective (absolutely pure) if ExtR1N,M=0 for all finitely presented R-modules N. Let X be a class of left R-modules. Mao and Ding in [2] introduced the concept of X-injective modules (see Definition 6). Selvaraj et al. developed X-injective and X-flat R-modules and studied covers and envelopes of modules with Goresntein properties in [3–5].A pairA,B is a cotorsion theory (see Definition 4). In our article, A,B is a cotorsion theory generated by the class X [6], that is, A=⊥X⊥ and B=X⊥. If R is a hereditary ring (that is, every ideal is projective), then A is closed under submodules and containing all injective R-modules in the cotorsion theory A,B. In this case, results are trivial. For this reason, we introduced X-hereditary ring (see Definition 16), and we restrict the setting to X⊥-hereditary rings.The notions of (pre)covers and (pre)envelopes of modules were introduced by Enochs in [7] and, independently, by Auslander and Smalø in [8]. Since then, the existence and the properties of (pre)covers and (pre)envelopes relative to certain submodule categories have been studied widely. The theory of (pre)covers and (pre)envelopes, which play an important role in homological algebra and representation theory of algebras, is now one of the main research topics in relative homological algebra.Salce introduced the notion of a cotorsion theory in [9]. Enochs showed the important fact that closed and complete cotorsion pairs provide minimal versions of covers and envelopes. Eklof and Trlifaj [10] proved that a cotorsion pair A,B is complete when it is cogenerated by a set. Consequently, many classical cotorsion pairs are complete. In this way, Bican et al. [11] showed that every module has a flat cover over an arbitrary ring. These results motivate us to define X-projective R-modules (see Definition 14) and ⊥X⊥-pure projective R-modules (see Definition 21), and we prove the existence of X⊥-projective cover and X-injective envelope. In particular, we denote PI by the class of all pure injective R-modules, and we prove the following main result.Theorem 1.
LetR be a X⊥-hereditary ring and X⊥⊆PI. Then, every R-module M has a X⊥-projective cover and an X-injective envelope.Self-injective rings were introduced by Johnson and Wong in [12]. A ring R is said to be self injective if R over itself is an injective R-module. In this paper, we introduce self-X-injective ring (see Definition 27). Recall that an R-module M is called reduced [13] if it has no nonzero injective submodules. An R-module M is said to be coreduced [14] if it has no nonzero projective quotient modules. Mao and Ding [15] proved that an FI-injective R-module decomposes into an injective and a reduced FI-injective R-module over a coherent ring. Similarly, we prove the following result.Theorem 2.
LetR be a self X-injective and X⊥-hereditary ring and X⊥⊆PI. Then, an R-module M is X⊥-projective if and only if M is a direct sum of a projective R-module and a coreduced X⊥-projective R-module.Pinzon [16] proved that every module has an fp-injective cover over a coherent ring. We prove the following result that provides the existence of W-injective cover, where W is the class of all pure projective R-modules and S the class of all small−Ω1-R-modules (see Definition 41).Theorem 3.
LetR be a coherent ring and W⊆S. Then, every R-module has a W-injective cover.This paper is organized as follows: In Section2, we recall some notions that are necessary for our proofs of the main results of this paper.In Section3, we investigate the notions of X-injective and X-flat modules and give some characterizations of these classes of modules.In Section4, we introduce X⊥-hereditary ring. Further, we investigate X⊥-projective module and give some characterizations. Further, we prove Theorems 1 and 2.In Section5, we prove that if M is a submodule of an X-injective R-module A, then, i:M⟶A is a special X-injective envelope of M if and only if A is an X⊥-projective essential extension of M.In the last section, we assume thatW is the class of all pure projective R-modules and we prove that every R-module has a W-injective preenvelope. Finally, we prove the main theorem of this section Theorem 3.


## 2. Preliminaries

In this section, we recall some known definitions and some terminology that will be used in the rest of the paper.Given a classC of left R-modules, we write
(1)C⊥=N∈R−ModExtR1M,N=0∀M∈C,⊥C=N∈R−ModExtR1N,M=0∀M∈C.Following [7], we say that a map f∈HomRC,M with C∈C is a C-precover of M, if the group homomorphism HomRC′,f:HomRC′,C⟶HomRC′,M is surjective for each C′∈C. A C-precover f∈HomRC,M of M is called a C-cover of M if f is right minimal, that is, if fg=f implies that g is an automorphism for each g∈EndRC. C⊆R−Mod is a precovering class (resp., covering class) provided that each module has a C-precover (resp., C-cover). Dually, we have the definition of C-preenvelope (resp., C-envelope).AC-precover f of M is said to be special [6] if f is an epimorphism and kerf∈C⊥.AC-preenvelope f of M is said to be special [6] if f is a monomorphism and cokerf∈⊥C.AC-envelope ϕ:M⟶C is said to have the unique mapping property [17] if for any homomorphism f:M⟶C′ with C′∈C, there is a unique homomorphism g:C⟶C′ such that gϕ=f.A module is said to bepure projective [18] if it is projective with respect to pure exact sequence.AnR-module M is called super finitely presented ([19]) if there exist a projective resolution with each projective R-module is finitely generated.Following [20, 21], an R-module M is called weak injective if ExtR1N,M=0 for every super finitely presented R-module N. A right R-module M is called weak flat if Tor1RM,N=0 for every super finitely presented R-module N.A classC of left R-modules is said to be injectively resolving [6] if C contains all injective modules and if given an exact sequence of left R-modules
(2)0⟶A⟶B⟶C⟶0.C∈C whenever A,B∈C.Definition 4.
A pairC=A,B of classes of modules is called a cotorsion theory [6] if A=⊥B and B=A⊥.(1)
A cotorsion theoryA,B is said to be perfect [22] if every module has an A-cover and a B-envelopeDefinition 5.
A cotorsion theoryA,B is said to have enough injectives [13] if for every R-module M there is an exact sequence 0⟶M⟶B⟶A⟶0 with B∈B and A∈A.For anR-module M, fdM denotes the flat dimension of M and idM denotes the injective dimension of M.TheX⊥-coresolution dimension of M, denoted by cores.dimX⊥M, is defined to be the smallest nonnegative integer n such that ExtRn+1A,M=0 for all R-modules A∈X (if no such n exists, set cores.dimX⊥M=∞), and cores.dimX⊥R is defined as supcores.dimX⊥MM∈R−Mod.We denote byℤ the ring of all integers and by ℚ the field of all rational numbers. For a left R-module M, we denote by M+=HomℤM,ℚ/ℤ the character module of M.I0 denotes the class of all injective left R-modules.For unexplained terminology, we refer to [23, 24].


## 3.X-Injective and X-Flat Modules

In this section, we assume thatX is any class in R−Mod. We begin with the following definition:Definition 6 (see [2]).
A leftR-module M is called X-injective if ExtR1X,M=0 for all left R-modules X∈X. A right R-module N is said to be X-flat if Tor1RN,X=0 for all left R-modules X∈X.
We denote byX⊥ the class of all X-injective modules and ⊥X⊥ the class of all X⊥-projective R-modules and, further, XΤ the class of all X-flat R-modules.Example 7.
(1)
IfX=R−Mod, then, X⊥ is the class of all injective R-modules and XΤ is the class of all flat right R-modules.(2)
IfX is the class of all finitely presented R-modules, then, X⊥ is the class of all fp-injective R-modules.(3)
IfX is the class of all left super finitely presented R-module, then, X⊥ is the class of all weak injective R-modules and XΤ is the class of all weak flat right R-modules.Proposition 8.
LetM be an R-module. Then, the following are true:
(1)
LetI0⊆X. Then, M is injective if and only if M is X-injective and idM≤1.(2)
M is injective if and only if M is X⊥-injective and cores.dimX⊥M≤1.Proof.
(1)
The direct implication is clear. Conversely, letM be X-injective and idM≤1. For any R-module N, consider an exact sequence 0⟶N⟶EN⟶L⟶0 with EN an injective envelope of N. We have an exact sequence(3)⋯⟶ExtR1EN,M⟶ExtR1N,M⟶ExtR2L,M⟶⋯
SinceidM≤1, ExtR2L,M=0, and hence, M is injective.
(2)
The direct implication is clear by the definition of an injective module. Conversely, letM be X⊥-injective and cores.dimX⊥M≤1. Consider an exact sequence 0⟶M⟶EM⟶L⟶0 with EM an injective envelope of M. For any R-module X∈X, we have an exact sequence ⋯⟶ExtR1X,EM⟶ExtR1X,L⟶ExtR2X,M⟶⋯. Since cores.dimX⊥M≤1, ExtR2X,M=0, and hence, L is X-injective. Therefore, ExtR1L,M=0, so that the exact sequence is split. It follows that M is a direct summand of E, as desired.We now give some of the characterizations ofX-injective module:Proposition 9.
LetI0⊆X. The following are equivalent for a left R-module M:(1)
M is X-injective(2)
For every exact sequence0⟶M⟶E⟶L⟶0, with E∈X,E⟶L is an X-precover of L(3)
M is a kernel of an X-precover f:A⟶B with A an injective module(4)
M is injective with respect to every exact sequence 0⟶A⟶B⟶C⟶0, with C∈X.Proof.
1⇒2. Consider an exact sequence
(4)0⟶M⟶E⟶L⟶0,whereE∈X. Then, by hypothesis, HomRE′,E⟶HomRE′,L is surjective for all left R-modules E′∈X, as desired.
2⇒3. Let EM be an injective hull of M and consider the exact sequence 0⟶M⟶EM⟶fEM/M⟶0. Since EM is injective, it belongs to X. So assertions 3holds.
3⇒1. Let M be a kernel of an X-precover f:A⟶B with A an injective module. Then, we have an exact sequence 0⟶M⟶A⟶A/M⟶0. Therefore, for any left R-module N∈X, the sequence HomRN,A⟶HomRN,A/M⟶ExtR1N,M⟶0 is exact. By hypothesis, HomRN,A⟶HomRN,A/M is surjective. Thus, ExtR1N,M=0, and hence (1) follows
1⇒4. Consider an exact sequence
(5)0⟶A⟶B⟶C⟶0,where C∈X. Then, HomRB,M⟶HomRA,M is surjective, as desired.
4⇒1. For each left R-module N∈X, there exists a short exact sequence 0⟶K⟶P⟶N⟶0 with P a projective R-module, which induces an exact sequence HomRP,M⟶HomRK,M⟶ExtR1N,M⟶0. By hypothesis, HomRP,M⟶HomRK,M⟶0 is exact. Thus, ExtR1N,M=0, and hence (1) follows.The following note is useful for understanding the notations in Examples11 and 15.Note 10.
From Introduction,B=X⊥. We get a new cotorsion theory ⊥B⊥,B⊥ generated by the class B.Example 11.
LetR,m be a commutative Noetherian and complete local domain. Assume that the depthR≥2 and cores.dimBR≤1. Then, R/m⊕ER is an B⊥-injective R-module, where B is the class of all X-injective R-modules.Proof.
Consider the residue fieldk=R/m and an exact sequence 0⟶k⟶Ek⟶ϕEk/k⟶0. If G is an R-module, the sequence HomRG,Ek⟶HomRG,Ek/k⟶ExtR1G,k⟶0 is exact. By ([25], p 43), ϕ is an injective cover of Ek/k. Since cores.dimBR≤1, then, the class of all injective R-modules and the class of all B-injective R-modules are equal, that is, I0=B⊥. Clearly, ϕ is an B-injective cover of Ek/k. Thus, HomRG′,Ek⟶HomRG′,Ek/k is surjective for every B-injective R-module G′. We get ExtR1G′,k=0 for every B-injective R-module G′, and hence, k is B⊥-injective. On the other hand, ER is injective and so B⊥-injective. Therefore, k⊕ER is B⊥-injective.We now give some characterizations ofX-flat module:Proposition 12.
The following are equivalent for a rightR-module M:
(1)
M is X-flat(2)
For every exact sequence0⟶A⟶B⟶C⟶0 with C∈X, the functor M⊗R− preserves the exactness(3)
ExtR1M,G+=0 for all G∈X(4)
M+ is X-injectiveProof.
1⇒2. Consider an exact sequence 0⟶A⟶B⟶C⟶0 with C∈X. Since M is X-flat, Tor1RM,C=0. Hence, the functor M⊗− preserves the exactness.
2⇒1. Let G∈X. Then, there exists a short sequence 0⟶K⟶F⟶G⟶0 with F a projective module, which induces an exact sequence 0⟶Tor1RM,G⟶M⊗K⟶M⊗F⟶M⊗G⟶0. By hypothesis, Tor1RM,G=0. Thus, M is X-flat.
1⇔3.It follows from the natural isomorphism ([26], p 34) Tor1RM,G+≅ExtR1M,G+.
1⇔4.It follows from the natural isomorphism ([27], VI 5.1) ExtR1G,M+≅Tor1RM,G+.Proposition 13.
LetR be a coherent ring. Then, a right R-module N is flat if and only if N is X-flat and fdN≤1.Proof.
The “only if” part is trivial. Conversely, suppose thatN is X-flat. By Proposition 12, N+ is X-injective. By ([28], Theorem 2.1), fdN=idN+. Then cores.dimX⊥N+≤1 since cores.dimX⊥N+≤idN+. By Proposition 8, N+ is injective and hence N is flat.


## 4.X⊥-Projective Cover and X-Injective Envelope

Now, we introduceX-projective modules.Definition 14.
AnR-module M is called X-projective if ExtR1M,X=0 for all R-modules X∈X.Example 15.
LetR,m be a complete local ring and B be the class of all X-injective R-modules. Assume that the cores.dimBR≤1. Then Ek/k is B⊥-projective, where k=R/m is the residue field and EM is an B-injective envelope of k.Proof.
By Note10, we get a cotorsion theory ⊥B⊥,B⊥ generated by the class B. Consider an exact sequence 0⟶k⟶Ek⟶Ek/k⟶0, where Ek is an injective envelope of k. Since cores.dimBR≤1, then, the class of all injective R-modules and the class of all B-injective R-modules coincide, that is, I0=B⊥. It follows that Ek is an B-injective envelope of k. Since the class of all B⊥-projective modules is closed under extensions, then by Lemma 2.1.2 in [25], Ek/k is B⊥-projective.We now introduce Definition16.Definition 16.
A ringR is called left X-hereditary if every left ideal of R is X-projective.Remark 17.
IfX is the class of injective left R-modules, then, every ring is X-hereditary. It is also easy to see that a ring is left hereditary if and only if R is X-hereditary for every class X of left R-modules.Given a classX of left R-modules, we denote by X⊥2 the class
(6)N∈R−ModExtR2X,N=0,foreveryX∈X.Example 18.
LetR be a commutative Noetherian ring. If X=R/p:p∈SpecR, then R is a X⊥2-hereditary ring.Proof.
LetI be an ideal of R. We claim that I is X⊥2-projective. By hypothesis, ExtR2R/p,G=0 for all p∈SpecR and for all G∈X⊥2. It follows that idG≤1. Thus, ExtR2R/I,G=0 for all ideals I of R and for all G∈X⊥2. Consider an exact sequence 0⟶I⟶R⟶R/I⟶0. So ExtR1I,G=0 for all G∈X⊥2. Hence, I is X⊥2-projective, as desired.
Note that anR-module M is X⊥-projective if ExtR1M,U=0 for all R-modules U∈X⊥. Clearly, ⊥X⊥,X⊥ is a cotorsion theory.Proposition 19.
A ringR is left X⊥-hereditary if and only if every submodule of a X⊥-projective left R-module is X⊥-projective.Proof.
LetR be left X⊥-hereditary and I be a left ideal of R. Then, there is an exact sequence 0⟶I⟶R⟶R/I⟶0. By hypothesis, ExtR2R/I,U=0 for any X-injective left R-module U. Thus, idU≤1. Let G be a submodule of an X⊥-projective left R-module H. Then, for any X-injective left R-module U, the sequence
(7)⋯⟶ExtR1H,U⟶ExtR1G,U⟶ExtR2HG,U⟶⋯is exact. Thus, ExtR1G,U=0 since H is X⊥-projective and idU≤1. The reverse implication is clear.
In general,⊥X⊥ is not closed under pure submodules; for example, if F is the class of flat modules, then ⊥F⊥=F and this class is not closed under pure submodules in general. As an easy consequence of Proposition 19, we have that the class ⊥X⊥ is closed under pure submodules over a X⊥-hereditary ring.Definition 20 (see [22]).
LetK be a class of R-modules. Then, K is said to be Kaplansky class if there exists a cardinal ℵ such that for every M∈K and for each x∈M, there exists a submodule K of M such that x∈K⊆M and K,M/K∈K, and CardK≤ℵ.Definition 21.
AnR-module G is called ⊥X⊥-pure injective if f:A′⟶G extends to a homomorphism g:A⟶G for all A′,A∈⊥X⊥, where A′⊆⋆A (that is, A′ is a pure submodule of A). We denote by B′ the class of all ⊥X⊥-pure projective modules.Remark 22.
The class of all pure injectiveR-modules is contained in the class of all ⊥X⊥-pure injective R-modules.Example 23.
(1)
Clearly,R−Mod,I0 is a cotorsion theory. Then, the class of all R−Mod-pure injective R-modules is the class of all pure injective R-modules.(2)
IfF is the class of all flat R-modules and C is the class of all cotorsion R-modules, then, F,C is a cotorsion theory. Then, the class of all F-pure injective R-modules is the class of all cotorsion R-modules.
IfR is Noetherian and R∈I0, then, the class of all I0-pure injective R-modules is I0⊥.Proof.
(1)
Straightforward(2)
LetC be a F-pure injective R-module. We show that C is a cotorsion R-module. For any flat R-module F, there is a pure exact sequence 0⟶K⟶P⟶F⟶0, where P is a projective R-module. Then, we get the following exact sequence:(8)HomRP,C⟶HomRK,C⟶ExtR1F,C⟶0
It follows thatExtR1F,C=0 because HomRP,C⟶HomRK,C is surjective. Hence, C is cotorsion. Conversely, let C be a cotorsion R-module. Suppose the following sequence:
(9)0⟶A⟶F⟶AF⟶0,where A,F∈F is pure exact. Then, we get the exact sequence HomRF,C⟶HomRA,C⟶ExtR1F/A,C⟶0. It follows that ExtR1F/A,C=0 since F/A is flat. Hence C, is F-pure injective.
(3)
By Exercises 6 in [13], I0,I0⊥ is a cotorsion theory. Let G be an I0-pure injective R-module. We show that G∈I0⊥. For any R-module I0∈I0, consider the following exact sequence:(10)E•:0⟶K⟶P⟶I0⟶0,where P is projective. By hypothesis, P is injective. Since the cotorsion theory I0,I0⊥ has enough injectives, K is injective, that is K∈I0. It follows that the above sequence E• is pure exact. Then, we get the exact sequence HomRP,G⟶HomRK,G⟶ExtR1I0,G⟶0. Thus, ExtR1I0,G=0 since G is I0-pure injective. Hence, G∈I0⊥. Conversely, Let G be an I0-injective R-module. Suppose the following exact sequence:
(11)0⟶E1⟶E2⟶E2E1⟶0,where E1,E2∈I0 is pure exact. It follows that E2/E1∈I0 since the cotrosion theory I0,I0⊥ has enough injectives. Then, HomRE2,G⟶HomRE1,G is surjective. Hence, G is I0-pure injective.Proposition 24.
Let be aX⊥-hereditary ring and X⊥⊆B′. Then, ⊥X⊥ is a Kaplansky class.Proof.
LetM∈⊥X⊥ and x∈M. Consider the inclusion <x>⟶M, and we get by Lemma 5.3.12 in [13], a cardinal ℵα and a pure submodule F⊆M such that <x>⊆F and CardF≤ℵα. We get the pure exact sequence
(12)0⟶F⟶M⟶MF⟶0.
By Proposition19, F is X⊥-projective. It follows that
(13)HomRM,G⟶HomRF,G⟶ExtR1MF,G⟶0,for any X-injective R-module G. This implies that HomRM,G⟶HomRF,G is surjective by hypothesis. Thus, ExtR1M/F,G=0. Hence, we proved the proposition.Theorem 25.
LetR be a X⊥-hereditary ring and X⊥⊆B′. If the class of all X-projective R-modules is closed under direct limits, then, every R-module M has a X⊥-projective cover and an X-injective envelope.Proof.
By Proposition24, ⊥X⊥ is a Kaplansky class. Since all projective modules are X⊥-projective, ⊥X⊥ contains the projective modules. Clearly, ⊥X⊥ is closed under extensions. By hypothesis, ⊥X⊥ is closed under direct limits. Then by Theorem 2.9 in [22], ⊥X⊥,X⊥ is a perfect cotorsion theory. Hence, by Definition 4, every module has a ⊥X⊥-cover and a X⊥-envelope.Note that ifX⊥⊆PI, then the class of all X⊥-projective R-modules is closed under direct limits by Lemma 3.3.4 in [6]. Then, by Theorem 25 and Remark 22, we have the following.Theorem 26.
LetR be a X⊥-hereditary ring and X⊥⊆PI. Then, every R-module M has a X⊥-projective cover and an X-injective envelope.Now we introduce a self-X-injective ring.Definition 27.
A ringR is said to be self X-injective if R over itself is an X-injective module.Example 28.
(1)
If the classX is R−Mod, then R is a self injective ring(2)
IfX is the class of all finitely presented R-modules, then R is a self fp-injective ring(3)
IfX is the class of all flat R-modules, then R is a cotorsion ringWe now give some characterizations ofX⊥-projective module:Proposition 29.
LetR be a self X-injective ring, and let M be an R-module. Then, the following conditions are equivalent:
(1)
M is X⊥-projective(2)
M is projective with respect to every exact sequence 0⟶A⟶B⟶C⟶0, with A an X-injective R-module(3)
For every exact sequence0⟶K⟶P⟶M⟶0, where P is X-injective, K⟶P is an X-injective preenvelope of K(4)
M is cokernel of an X-injective preenvelope K⟶P with P is a projective R-moduleProof.
1⇒2. Let 0⟶A⟶B⟶C⟶0 be an exact sequence, where A is X-injective. Then, by hypothesis, HomRM,B⟶HomRM,C is surjective.
2⇒1. Let N be an X-injective R-module. Then, there is a sequence 0⟶N⟶E⟶L⟶0 with E an injective envelope of N. By (2), HomRM,E⟶HomRM,L is surjective. Thus, ExtR1M,N=0, as desired.
1⇒3. Clearly, ExtR1M,F′=0 for all X-injective F′. Hence, we have an exact sequence HomRP,F′⟶HomRK,F′⟶0, where P is projective.
3⇒4. Consider an exact sequence 0⟶K⟶P⟶M⟶0, where P projective. Since R is self-X-injective, every projective module is X-injective. Hence, P is X-injective. Then, by hypothesis, K⟶P is an X-injective preenvelope.
4⇒1. By hypothesis, there is an exact sequence 0⟶K⟶P⟶M⟶0, where K⟶P is an X-injective preenvelope with P projective. It gives rise to the exactness of HomRP,N⟶HomRK,N⟶ExtR1M,N⟶0 for each X-injective R-module N. Since R is self X-injective, HomRP,N⟶HomRK,N is surjective. Hence, ExtR1M,N=0, as desired.Proposition 30.
LetR be a self X-injective and X⊥-hereditary ring and X⊥⊆PI. Then the following are equivalent for an R-module M:
(1)
M is coreduced X⊥-projective(2)
M is a cokernel of an X-injective envelope K⟶P with P is a projective R-moduleProof.
1⇒2. Consider an exact sequence 0⟶K⟶fP⟶gM⟶0 with P a projective module. Since R is self X-injective, P is X-injective. By Theorem 26, the natural map f:K⟶P is an X-injective preenvelope of K. By hypothesis, K has an X-injective envelope α:K⟶P′. Then, there exist β:P′⟶P and β′:P⟶P′ such that α=β′f and f=βα. Hence, α=ββ′α. It follows that ββ′ is an isomorphism, P=imβ⊕kerβ′. Note that imf⊆imβ, and so P/imf⟶P/imβ⟶0 is exact. But M is coreduced and P/imf≅M, and hence, P/imβ=0, that is, P=imβ. So β is an isomorphism, and hence, f:K⟶P is an X-injective envelope of K.
2⇒1. By Proposition 29, M is X⊥-projective and M is coreduced by Lemma 3.7 in [17].We are now to prove the main result of this section.Theorem 31.
LetR be a self X-injective and X⊥-hereditary ring and X⊥⊆PI. Then, an R-module M is X⊥-projective if and only if M is a direct sum of a projective R-module and a coreduced X⊥-projective R-module.Proof.
The “if” part is clear.
“Only if” part. LetM be a X⊥-projective R-module. By Proposition 29, we have an exact sequence 0⟶K⟶P⟶M⟶0 with P a projective module, where K⟶P is an X-injective preenvelope of K. By Theorem 26, K has an X-injective envelope f∈HomRK,P′ with P′ an X-injective R-module. Then, we have the following diagram (Figure 1):Figure 1
Commutative diagram with exact rows.Note thatβα is an isomorphism, and so P=kerβ⊕imα. Since imα≅P′, P′ and kerβ are projective. Therefore, P′/imf is a coreduced X⊥-projective module by Proposition 30. By the Five Lemma, σϕ is an isomorphism. Hence, we have M=imϕ⊕kerσ, where imϕ≅P′/imf. In addition, we get the commutative diagram (Figure 2).Figure 2
Pullback diagram.Hence,kerσ≅kerβ.


## 5. Some Relation betweenX⊥-Projective and X-Injective Modules

In this section, we deals withX-injective envelope of a module and X⊥-projective module for any class X in R−Mod.Theorem 32.
Letϕ:M⟶A be an X-injective envelope. Then, L=A/ϕM is X⊥-projective, and hence, A is X⊥-projective whenever M is X⊥-projective.Proof.
It follows from Lemma 2.1.2 in [25].Theorem 33.
Let0⟶M⟶A⟶D⟶0 be a minimal generator of all X⊥-projective extensions of M. Then, A is an X-injective envelope of M.Proof.
It follows from Theorem 2.2.1 in [25].LetM be a submodule of a module A. Then, A is called a X⊥-projective extension of a submodule M if A/M is X⊥-projective.Recall that among allX⊥-projective extensions of M, we call one of them 0⟶M⟶A⟶D⟶0 a generator for Ext⊥X⊥,M (or a generator for all X⊥-projective extensions of M) if for any X⊥-projective extension 0⟶M⟶A′⟶D′⟶0 of M, then there is a diagram (Figure 3).Figure 3
Commutative diagram.Furthermore, a generator0⟶M⟶A⟶D⟶0 is called minimal if for all the vertical maps are isomorphisms whenever A′,D′ are replaced by A, D, respectively.Theorem 34.
LetR be a X⊥-hereditary ring and X⊥⊆PI. Then, for an R-module M, there must be a minimal generator whenever Ext⊥X⊥,M has a generator.Proof.
It follows from Theorem 2.2.2 in [25].Theorem 35.
Suppose that anR-module M has an X-injective envelope. Let M be a submodule of an X-injective R-module L. Then, the following are equivalent:
(1)
i:M⟶L is a special X-injective envelope
L/M is X⊥-projective, and there are no direct summands L1 of L with L1≠L and M⊆L1(2)
L/M is X⊥-projective, and for any epimorphism α:L/M⟶N such that απ is split, N=0, where π:L⟶L/M is the canonical map(3)
L/M is X⊥-projective, and any endomorphism γ of L such that γi=i is a monomorphism(4)
L/M is X⊥-projective, and there is no nonzero submodule N of L such that M∩N=0 and L=M⊕N is X⊥-projectiveProof.
1⇔2. It follows from Corollary ~1.2.3 in [25] and Theorem 32.
2⇒3. Since απ is split, there is a monomorphism β:N⟶L such that L=kerαπ⊕βN. Note that M⊆kerαπ, and so L=kerαπ by (2). Thus, βN=0, and hence, N=0.
3⇒2. If L=L1⊕N with M⊆L1. Let p:L⟶N be a canonical projection. Then, there is an epimorphism α:L/M⟶N such that απ=p. Thus, N=0 by hypothsis, and hence, L=L1, as required.
1⇒4. By Wakamatsu’s Lemma ([13], Proposition ~7.2.4), L/M is X⊥-projective. Since γi=i and i is monomorphism, γ is monomorphism.
4⇒1. Since L/M is X⊥-projective, i is a special X-injective preenvelope. Let ψ:M⟶X⊥M be an X-injective envelope of M. Then, there exist μ:L⟶X⊥M and ν:X⊥M⟶L such that μi=ψ and νψ=i. Hence μνψ=ψ and i=νμi. Thus, μν is an isomorphism, and so μ is epic. In addition, by (4), νμ is monic, and hence, μ is monic. Therefore, μ is an isomorphism, and hence, i is an X-injective envelope of M.
1⇒5. It is obvious that L/M is X⊥-projective. Suppose there is a nonzero submodule N⊆L such that M∩N=0 and L=M⊕N is X⊥-projective. Let π:L⟶L/N be a canonical map. Since L/N⊕M is X⊥-projective and L is X-injective, there is a β:L/N⟶L in the following diagram (Figure 4)Figure 4
Commutative diagram with exact rows.Hence,βπi=i. Note that i is an envelope, and so βπ is an isomorphism, whence π is an isomorphism. But this is impossible since πN=0.5⇒1. Let ψM:M⟶X⊥M be an X-injective envelope of M. Since L/M is X⊥-projective, i is a special X-injective preenvelope. Thus, we have the following diagram shown in Figure 5.Figure 5
Commutative diagram with exact rows.That is,fψM=i,gi=ψM. So gfψM=ψM. Note that ψM is an X-injective envelope, and hence, gf is an isomorphism. Without loss of generality, we may assume gf=1. Write α=ϕg:L⟶Q. It is clear that α is epic and M∩kerg=0. We show that M⊕kerg=kerα. Clearly, M⊕kerg⊆kerα. Let x∈kerα. Then, αx=ϕgx=0. It follows that gx=ψMm for some m∈M, and hence, fgx=fψMm=m,gx=gfgx=gm. Thus, x∈M⊕kerg, and so kerα⊆M⊕kerg, as desired. Consequently, L=M⊕kerg=L/kerα≅Q is X⊥-projective by Wakamatsu’s Lemma. Thus, kerg=0 by hypothesis, and hence, g is an isomorphism. So i:M⟶L is an X-injective envelope.Theorem 36.
LetR be a X⊥-hereditary ring and X⊥⊆PI. If M is a submodule of an X-injective R-module A, then, the following are equivalent:
(1)
i:M⟶A is a special X-injective envelope of M
A is a X⊥-projective essential extension of M.Proof.
1⇒2. It follows by Proposition 35.
2⇒1. By hypothesis, we have an exact sequence: 0⟶M⟶A⟶L⟶0 with A an X-injective module and D an X⊥-projective module. This sequence is a generator of all X⊥-projective extensions of M. By Theorems 33 and 34, we have an X⊥-projective extension sequence of M0⟶M⟶A′⟶L′⟶0 which gives an X-injective envelope of M. Then, we have the diagram shown in Figure 6.Figure 6
Commutative diagram.It is easy to see thatA=fA′⊕kerg. We claim that kerg=0. Since M=fα′M⊆fA′, kerg∩M=0. We define the following homomorphism ψ:A/M⊕kerg⟶L′, a+M⊕kerg↦α′ga. Obviously, ψ is well defined. By diagram chasing, we see that ψ is injective. But both g and β′ are surjective, so is ψ. Therefore, kerg is X⊥-projective essential extension of A/kerg. This contradicts the hypothesis that A is X⊥-projective essential extension of M. This implies that kerg=0 and so f is an isomorphism.


## 6.W-Injective Cover

In this section, we assumeW is the class of all pure projective R-modules and we prove that all modules have W-injective covers.Proposition 37.
The classW⊥ of all W-injective modules is closed under pure submodules.Proof.
LetA be a pure submodule of a W-injective module M. Then there is a pure exact sequence 0⟶A⟶M⟶M/A⟶0 and a functor HomRX,− preseves this sequence is exact whenever X∈W. This implies that the sequence 0⟶HomRW,A⟶HomRW,M⟶HomRW,M/A⟶ExtR1W,A⟶0 is also exact for all W∈W. It follows that ExtR1W,A=0 for all W∈W, as desired.Theorem 38.
EveryR-module has a W-injective preenvelope.Proof.
LetM be an R-module. By Lemma 5.3.12 in [13], there is a cardinal number ℵα such that for any R-homomorphism ϕ:M⟶G with G a W-injective R-module, there exists a pure submodule A of G such that A≤ℵα and ϕM⊂A. Clearly, W⊥ is closed under direct products and by Proposition 37A is W-injective. Hence, the theorem follows Proposition 6.2.1 in [13].Proposition 39.
The classW⊥ of all W-injective modules is injectively resolving.Proof.
Let0⟶M1⟶ϕM2⟶ψM3⟶0 be an exact sequence of left R-modules with M1,M2∈W⊥. Let G∈W⊥. By Theorem 38, every module has a W⊥-preenvelope. By Lemma 1.9 in [29], G has a special W⊥-preenvelope. By Lemma 2.2.6 in [6], G has a special ⊥W⊥-precover. Then, there exists an exact sequence 0⟶K⟶A⟶G⟶0 with A∈⊥W⊥ and K∈W⊥. We prove that M3 is W-injection, i.e., to prove that ExtR1G,M3=0. For this, it suffices to extend any α∈HomRK,M3 to an element of HomRA,M3. Clearly, K has ⊥W⊥-precover,
(14)0⟶K′⟶fA′⟶gK⟶0,where K, K′∈W⊥, and A′∈⊥W⊥. As the class W⊥ is closed under extensions, A′∈W⊥. Since α∘g:A′⟶M3 with A′∈W⊥ and M1 a W-injective module, then there exists β:A′⟶M2 such that ψ∘β=α∘g. That is, we have the diagram shown in Figure 7.Figure 7
Commutative diagram.Now, we defineβ↾imϕ:A′⟶imϕ, where ↾ is a restriction map. Then, there exists γ:K′⟶M1 such that β↾imϕfK′=ϕγK′. Hence, we have the diagram shown in Figure 8.Figure 8
Commutative diagram.TheW-injectivity of M1 yields a homomorphism γ1:A′⟶M1 such that γ=γ1∘f. So for each k′∈K′, we get β∘fk′=ϕ∘γk′=ϕ∘γ1∘fk′. Then, there exists a map β1∈HomRK,M2 such that β=β1∘g, and we get α=ψ∘β1. Thus, we have the diagram shown in Figure 9.Figure 9
Commutative diagram.SinceM2 is W-injective, there exists ρ∈HomRA,M2 such that β1=ρ∘f. Thus, α=ψ∘β1=ψ∘ρ∘f, where ψ∘ρ∈HomRA,M3. Hence, M3 is W-injective.Proposition 40.
LetM be a W-injective R-module and A be a pure submodule M. Then, an R-module M/A is W-injective.Proof.
By Proposition39, W⊥ is injectively resolving. Let M∈W⊥ and A be a pure submodule of M. By Proposition 37, A is W-injective. From the short exact sequence 0⟶A⟶M⟶M/A⟶0, we get M/A is W-injective.The terminology of the Definition41 is used in [30].Definition 41 (see [31]).
AnR-module M is fp−Ω1small−Ω1 if there is a projective resolution
(15)P•:⋯⟶P2⟶P1⟶ϕP0⟶M⟶0,such that the first syzygy Ω1P•=imϕ is finitely presented (small). We denote by C the class of all fp−Ω1R-modules and S the class of all small−Ω1-modules.Theorem 42 (see [31], Theorem 2.3).
LetM be an R-module.
(1)
IfM is an fp−Ω1-module, then ExtR1M,− commutes with direct limits(2)
IfM is a small-fp−Ω1-module, then ExtR1M,− commutes with direct sums
The converses of these statements are true when anR-module M is finitely generated.We recall that anR-module M is pure projective if and only if M is a direct summand of a direct sum of finitely presented R-modules [32]. Warfield proved that every pure projective R-module is a direct sum of finitely presented modules over a Noetherian local ring in Corollary 4 in [33]. Further, Puninski and Rothmalar [18] proved that the general question of when all pure projective modules are direct sum of finitely presented R-modules. They proved this over a hereditary Noetherian ring ([18], Corollary 6.5). Thus, every pure projective R-module is finitely presented over a hereditary Noetherian ring. It follows that ExtR1A,− commutes with direct limits for all pure projective R-modules A. By Theorem 42, every pure projective R-module is fp−Ω1 over a hereditary Noetherian ring.The converse part of the Proposition43 is not necessarily true. It will be held immediately over a hereditary Noetherian ring by the argument of the previous paragraph. In this article, we proved the converse part of Proposition 43 over a Noetherian ring by Theorem 45; that is, W-injective R-modules are injective from ([34], Theorem 3).Proposition 43.
EveryW-injective R-module is absolutely pure.Proof.
Every finitely presentedR-module is pure projective, hence the proposition.Theorem 44.
LetR be a perfect ring. Then the following conditions are equivalent:
(1)
R is coherent(2)
The class ofW-injective R-modules is closed under direct limits and closed under direct sums(3)
Every absolutely pureR-module is injectiveProof.
1⇒2. It follows by Corollary 57 (2) and (3) in [30].
2⇒3. By Proposition 43 and Theorem 3.2 in [1], R is coherent. A perfect coherent ring R is Noetherian. Then, we get (3) immediately by Theorem 3 in [34].LetFP denote the class of all absolutely pure R-modules.Theorem 45.
The following conditions are equivalent:(i)
R is Noetherian(ii)
W-injective R-modules coincide with the injective R-modules(iii)
Absolutely pureR-modules coincide with injective R-modulesProof.
i⇒ii. By Proposition 43, every W-injective R-module is an absolutely pure R-module. It follows that every absolutely pure R-module is injective by Theorem 3 in [34]. Hence, every W-injective R-module is injective.
ii⇒iii. Clearly, I0⊆FP⊆W⊥. By (ii), W⊥⊆I0. Thus I0=FP
iii⇒i. It follows Theorem 3 in [34].By Theorem45, W-injective cover and injective cover for every R-module are coincide over a Noetherian ring. Therefore, we prove Theorem 46 and Proposition 47 without Noetherian condition.The following result establishes an analog version of Theorem 2.6 in [16].Theorem 46.
SupposeW⊆S. Then, every R-module M has a W-injective precover.Proof.
LetW be a set with CardW≤κ, where κ is the cardinal in [11] (Theorem 5). Denote PW the power set of W. We find all the binary operations ∗:G×G⟶G for each element G∈PW, and we get a new collection ∪G∈PWG,∗=G¯′. From G¯′, find all the scalar multiplications, which are functions from the cross product into itself. This remains a set ∪G∈PWG,∗,· which is denoted by G¯. Some collection of members of G¯ form a module, and we can get the class G of W-injective modules which is contained in the class W⊥. Clearly, I0 is contained in the class G. Since W⊆S, then by Theorem 42 (2) the direct sum of W-injective R-modules is W-injective. Hence ⊕N∈I0NHomRN,M is an W-injective R-module because every injective module is W-injective. We prove that ⊕N∈I0NHomRN,M⟶ϕM is a W-injective precover. That is, to show that if for any homomorphism N′⟶M with N′∈W⊥, then we have the following diagram (Figure 10):Figure 10
Commutative diagram.LetK be the kernel of the map N′⟶M, where N′ is sufficiently large. Then, N′/K is sufficiently small since N′/K≤M. By Bican et al.’s Theorem [11], K has a nonzero submodule L that is pure in N′. Therefore, L is W-injective by Proposition 37. This implies that N′/L is W-injective by Proposition 40. If N′/L is still sufficiently large, then repeat the process from the map N′/L⟶M. Since N′/K1 is sufficiently small, K1/L has a nonzero submodule L1/K1 that is pure in N′/L. Thus, N′/K1/L1/K1=N′/L1 is W-injective. But again, this may be too large. Then by continuing this process we get lim¯N′/Li is W-injective since W⊥ is closed under direct limits, and it is sufficiently small, namely lim¯N′/Li≤κ. Then the map N′⟶M can be factored through a W-injective module lim¯N′/Li. Let f∈HomRlim¯N′/Li,M. We define a map f¯:lim¯N′/Li⟶⊕NHomRN,M such that f¯n′+L′=nf,0,0,⋯, with nf=n. Clearly, f¯ is a linear map. Then, we have the diagram shown in Figure 11.Figure 11
Commutative diagram.Hence, we get the diagram shown in Figure12.Figure 12
Commutative diagram.Thus,⊕N∈I0NHomRN,M⟶ϕM is a W-injective precover.By Theorem42 (1), the class of all W-injective R-modules are closed under direct limits if W⊆C. But we prove this result over a coherent in Proposition 47.Proposition 47.
LetR be a coherent ring. Then, the class of all W-injective R-modules is closed under direct limits.Proof.
It follows Corollary 57(2) shown in [30].Theorem48 follows from Theorem 46 and Proposition 47 by Corollary 5.2.7 in [13].Theorem 48.
LetR be a coherent ring and W⊆S. Then, every R-module has a W-injective cover.



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