The class of weak w-projective modules is a precover

Abstract

Let R be a commutative ring with identity. Denote by wPw the class of weak w-projective R-modules and by wPw^⊥ the right orthogonal complement of wPw. It is shown that (wPw, wPw^⊥) is a hereditary and complete cotorsion theory, and so every R-module has a special weak w-projective precover. We also give some necessary and sufficient conditions for weak w-projective modules to be w-projective. Finally it is shown that when we discuss the existence of a weak w-projective cover of a module, it is enough to consider the w-envelope of the module.

Full text

Bull. Korean Math. Soc. 00 (20XX), No. 0, pp. 1–0
https://doi.org/10.4134/BKMS.b210155
pISSN: 1015-8634 / eISSN: 2234-3016
THE CLASS OF WEAK w-PROJECTIVE MODULES
IS A PRECOVER
Hwankoo Kim, Lei Qiao, and Fanggui Wang
Abstract. Let Rbe a commutative ring with identity. Denote by wPw
the class of weak w-projective R-modules and by wPw
⊥the right orthog-
onal complement of wPw. It is shown that (wPw,wPw
⊥) is a hereditary
and complete cotorsion theory, and so every R-module has a special weak
w-projective precover. We also give some necessary and sufficient con-
ditions for weak w-projective modules to be w-projective. Finally it is
shown that when we discuss the existence of a weak w-projective cover
of a module, it is enough to consider the w-envelope of the module.
1. Introduction
Throughout this paper Ris always a commutative ring with identity. We
first review some related concepts of w-modules. A finitely generated ideal J
of Ris called a GV-ideal if the homomorphism R→HomR(J, R) induced by
the inclusion map J →Ris an isomorphism. Denote by GV(R) the set of
GV-ideals of R. For any R-module N, set
torGV(R)(N) = {x∈N|there exists J∈GV(R) such that Jx = 0},
which is a submodule of N, called the total GV-torsion submodule of N. If
torGV(R)(N) = N, then Nis called a GV-torsion module; if torGV(R)(N) = 0,
then Nis called a GV-torsion-free module. A GV-torsion-free module Nis
called a w-module if Ext1
R(R/J, N ) = 0 for any J∈GV(R). Denote by Wthe
class of w-modules. The set of maximal w-ideals of Ris denoted by w-Max(R).
By [10, Theorem 6.2.15], an R-module Tis a GV-torsion module if and only if
Tm= 0 for any m∈w-Max(R).
We also need the concept of strong w-modules. An R-module Nis called
astrong w-module if Extk
R(T, N ) = 0 for any GV-torsion module Tand any
k>1. For a discussion of strong w-modules, please refer to [12]. Denote by
W∞the class of strong w-modules.
Received February 21, 2021; Accepted October 14, 2021.
2010 Mathematics Subject Classification. 13C10, 13D05, 13D07, 13D30.
Key words and phrases. Weak w-projective precover, w-operation (theory), cotorsion
theory.
c
20XX Korean Mathematical Society
1
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2 H. KIM, L. QIAO, AND F. WANG
Since the w-operation on an integral domain can establish the concept of w-
modules, which allows the w-operation to work in the category of modules, in
1997 the concepts of w-projective modules and w-flat modules over an integral
domain were first introduced [8]. In [4] the definition of w-flat modules was
extended to any commutative ring as follows. A module Mis called a w-flat
module if the functor M⊗ − preserves a w-exact sequence into a w-exact se-
quence. In [11] the concepts of the w-flat dimension (w-fd) of a module and the
w-weak global dimension (w-w.gl.dim) of a ring have been successively intro-
duced. Using the w-weak global dimension of a ring, a Pr¨ufer v-multiplication
domain (PVMD for short) can be characterized homologically as an integral
domain of w-w.gl.dim(R)61.
In 2015, the concept of w-projective modules was also extended to any com-
mutative ring [9]. Let Mbe an R-module. Set L(M) := (M/torGV(R)(M))w.
Then Mis called a w-projective module if Ext1
R(L(M), N ) is a GV-torsion mod-
ule for any w-module N. Denoted by Pwthe class of w-projective modules.
One can use the w-projective modules to introduce the w-projective ideals. One
hopes that some rings that used to be described by ideals can be described by
the w-projective modules. For example, in [11] it is proved that an integral
domain Ris a PVMD if and only if every finitely generated submodule of a
projective module is w-projective. As we all know, an integral domain Ris a
Dedekind domain if and only if each nonzero ideal is invertible; Ris a Krull
domain if and only if each nonzero ideal is w-invertible. Therefore, in the above
sense, Krull domains can actually be considered as w-Dedekind domains. But a
Dedekind domain is exactly an integral domain with global dimension at most
1, in other words, every submodule of a projective module is projective. In
[14], the authors can only prove that an integral domain Ris a Krull domain
if and only if every submodule of a finitely generated projective module is w-
projective. That is to say, the concept of w-projective modules cannot be used
to obtain a complete characterization of the Krull domains corresponding to
the Dedekind domains.
In order to give a complete homological characterization of Krull domains,
the concept of weak w-projective modules is introduced in [12] with the aid of
w-projective modules. Denote by RMthe category of all R-modules. Set
Pw†∞=N∈M



Nis GV-torsion-free and
Extk
R(M, N ) = 0 for any M∈ Pwand any k>1.
An R-module Mis called a weak w-projective module if Ext1
R(M, N )=0
for any N∈ Pw†∞. Denote by wPwthe class of weak w-projective modules.
In [12] the authors pointed out: Every w-pro jective module must be weak w-
projective. Conversely, every weak w-projective module of finite type and any
weak w-projective ideal of an integral domain are all w-projective. At the same
time, in [12] it is also given an example of a weak w-projective module over
a UFD, which is not w-projective. In [12] it is also introduced the concept
of the w-projective dimension (w-pd) of a module and the global w-projective
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WEAK w-PROJECTIVE MODULES 3
dimension (w-gl.dim) of a ring. With the help of the concepts of weak w-
projective modules and the global w-projective dimension of a ring, in [12]
the authors give a homological characterization of Krull domains: An integral
domain Ris a Krull domain if and only if every submodule of a projective
module is weak w-projective, equivalently, w-gl.dim(R)≤1.
Let Abe a class of modules, Mbe an R-module, A∈ A, and ϕ:A→Mbe
a homomorphism. Then (A, ϕ) is called an A-precover of Mif for any A0∈ A
and any homomorphism f:A0→M, the following diagram
A0
h
xx
f

Aϕ//M
is commutative, equivalently, for any A0∈ A, HomR(A0, A)ϕ∗
→HomR(A0, M )→
0 is an exact sequence. Let (A, ϕ) be an A-precover of a module M. When
A0=A,f=ϕ, and the above diagram is commutative, it is said that (A, ϕ) is
an A-cover of Mif his an isomorphism. If any R-module Mhas A-precover
(resp., cover), then we say that Ais a precover (resp, cover) class.
Let Sbe a class of modules. Set
⊥S:= {A∈M|Ext1
R(A, C) = 0 for any C∈ S}
and
S⊥:= {B∈M|Ext1
R(C, B) = 0 for any C∈ S},
are called the left orthogonal complement and the right orthogonal complement
of S, respectively [3]. Then obviously one has wPw=⊥(P†∞
w). In [1], the
authors introduced and studied the right orthogonal complement of the class
of w-flat modules. Also set
⊥∞S:= {A∈M|Extk
R(A, C) = 0 for any C∈ S and any k>1},
and
S⊥∞:= {B∈M|Extk
R(C, B) = 0 for any C∈ S and any k>1}.
In recent years, the cotorsion theory has received great attention from re-
searchers. Let A,Bbe two classes of modules. Then (A,B) is called a cotorsion
theory if B=A⊥and A=⊥B. In addition, (A,B) is called a hereditary co-
torsion theory if whenever 0 →A1→A→A2→0 is exact with A, A2∈ A,
one has A1∈ A. And (A,B) is called a complete cotorsion theory if for any
R-module M, there is an exact sequence 0 →K→A→M→0 with A∈ A
and K∈ B. When a formulated pair (A,B) of modules becomes a cotorsion
pair, the classical homology method can be used very smoothly to characterize
rings and modules. For the projective modules, a well-known theorem of Ka-
plansky states that a projective module over an arbitrary ring is a direct sum of
countably generated projective modules. In 2020, Wang and Qiao established
the w-version of Kaplansky’s theorem [13]: If Mis a w-projective w-module,
then Mhas a w-projective w-ℵ0-continuous ascending chain (see the definition
$KHDGRI3ULQW

4 H. KIM, L. QIAO, AND F. WANG
later). Using this result, this article obtains the main result: (wPw,wPw⊥) is
a hereditary and complete cotorsion theory, and so every module has a special
weak w-projective precover.
2. Basic results
Denoted by FT the class of GV-torsion-free modules. Let Sbe a class of
modules. Define:
S†:= S⊥∩ FT
={N∈M|Nis GV-torsion-free and Ext1
R(M, N ) = 0 for any M∈ S}
and
S†∞:= S⊥∞∩ FT
=N∈M



Nis GV-torsion-free and
Extk
R(M, N ) = 0 for any M∈ S and any k>1.
Set
GV(R)∗:= {R/J |J∈GV(R)}.
Obviously GV(R)∗is a set of modules.
Proposition 2.1. Let S,S1be classes of modules. Then:
(1) S ⊆ ⊥(S†∞)⊆⊥(S†).
(2) If S ⊆ S1, then S†
1⊆ S†and S†∞
1⊆ S†∞.
(3) (S ∪ S1)†=S†∩ S †
1.
Proof. These are obvious. 
For k>1, set
Wk:= {N∈ FT | Exti
R(R/J, N ) = 0 for any J∈GV(R) and any 1 6i6k}.
By convention, we set W0:= FT . A module Nis called a wk-module if
N∈ Wk. It is known that a GV-torsion-free module Nis a w-module if and
only if Ext1
R(C, N ) = 0 for any GV-torsion-module C([10, Theorem 6.2.7]).
Lemma 2.2. (1) If 16i6k, then Wk⊆ Wi.
(2) Wkis closed under extensions.
(3) Let N∈ Wk. Then N∈ Wk+1 if and only if Extk+1
R(M, N )=0for
any GV-torsion module M.
Proof. (1) and (2) are trivial. We will prove only (3). It is enough to show the
necessity. Assume that N∈ Wk+1 . If k= 0, then N∈ W1=W. Thus by
[10, Theorem 6.2.7], Ext1
R(M, N ) = 0 for any GV-torsion module M. Consider
the case k= 1. Let Mbe a GV-torsion module. Then for any x∈M, there
exists Ix∈GV(R) such that Ixx= 0. Set F:= L
x∈M
R/Ix. Then Fis a GV-
torsion module. Let exdenote the element in Fthat takes the value 1+Ixat the
component x, and the other components take the value 0. Define h:F→M
$KHDGRI3ULQW

WEAK w-PROJECTIVE MODULES 5
by h(ex) = x. Then his an epimorphism. Set A:= Ker(h). Then it follows
from the exact sequence 0 = Ext1
R(A, N )→Ext2
R(M, N )→Ext2
R(F, N )=0
that Ext2
R(M, N ) = 0. Now the assertion follows by induction. 
Proposition 2.3. The following are equivalent for a GV-torsion-free module
N.
(1) N∈ W∞.
(2) Exti
R(R/J, N )=0for any J∈GV(R)and any i>1.
Proof. (1) ⇒(2) This is trivial.
(2) ⇒(1) Let k>1 and set
W0
k:= N∈ FT 



Exti
R(M, N ) = 0 for any GV-torsion module M
and any 1 6i6k.
By Lemma 2.2, W0
k=Wk. Thus N∈T∞
k=1 W0
k=W∞.
Let Mand Nbe R-modules. A homomorphism f:M→Nis called a w-
monomorphism (resp., a w-epimorphism, a w-isomorphism) if fm:Mm→Nm
is a monomorphism (resp., an epimorphism, an isomorphism) for any maximal
w-ideal mof R. And Mis said to be w-isomorphic to Nprovided that there
exist an R-module Land two w-isomorphisms f:L→Mand g:L→N.
Theorem 2.4. Let Sbe a class of modules such that S ⊆ F T . Set A:= ⊥S.
Then the following are equivalent.
(1) Ais closed under w-isomorphisms.
(2) GV(R)∪GV(R)∗⊆ A.
(3) S ⊆ W2.
Proof. (1) ⇒(2) Let J∈GV(R). Since R∈ A and Jand Rare w-isomorphic,
it follows that J∈ A. Also since R/J and 0 are w-isomorphic, it follows that
R/J ∈ A.
(2) ⇒(3) Let N∈ S. Then Ext1
R(R/J, N ) = 0 and Ext1
R(J, N ) = 0 for any
J∈GV(R). Thus Nis a w2-module. Therefore S ⊆ W2.
(3) ⇒(1) Let f:M→M0be a w-isomorphism. By [10, Proposition 6.3.4],
there exist a module Band exact sequences 0 →A→M→B→0 and
0→B→M0→C→0, where Aand Care GV-torsion modules. If M∈ A,
then for any N∈ S it follows from the exact sequence 0 = HomR(A, N )→
Ext1
R(B, N )→Ext1
R(M, N ) = 0 that Ext1
R(B, N ) = 0. Again by the exact
sequence 0 = Ext1
R(C, N )→Ext1
R(M0, N )→Ext1
R(B, N ) = 0 it follows that
Ext1
R(M0, N ) = 0, that is, M0∈ A.
On the other hand, assume that M0∈ A. By Lemma 2.2, Ext2
R(C, N ) = 0.
By the exact sequence 0 = Ext1
R(M0, N )→Ext1
R(B, N )→Ext2
R(C, N ) = 0 it
follows that Ext1
R(B, N ) = 0. Also by the exact sequence 0 = Ext1
R(B, N )→
Ext1
R(M, N )→Ext1
R(A, N ) = 0, it follows that Ext1
R(M, N ) = 0, i.e., M∈ A.
Therefore Ais closed under w-isomorphisms. 
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6 H. KIM, L. QIAO, AND F. WANG
Corollary 2.5. Let Sbe a class of modules. Set A:= ⊥S. If S ⊆ W∞, then
Ais closed under w-isomorphisms.
Proof. This follows directly from Theorem 2.4 and the fact that W∞⊆ W2.
Example 2.6. (1) It is easy to see that (GV(R)∗)†=W.
(2) By Proposition 2.3, (GV(R)∗)†∞=W∞.
(3) By Theorem 2.4, (GV(R)∗∪GV(R))†=W2.
Proposition 2.7. Let Sbe a class of modules satisfying GV(R)∗⊆ S. Then:
(1) S†⊆ W and S†∞⊆ W∞.
(2) If GV(R)⊆ S, then S†⊆ W2.
Proof. This follows immediately from Example 2.6. 
3. The class of weak w-projective modules is a precover
Let Abe a class of modules and Mbe an R-module. If there is a continuous
ascending chain of submodules of M:
(3.1) 0 = M0⊆M1⊆ · · · ⊆ Mα⊆Mα+1 ⊆ · · · ⊆ Mλ=M
such that Mα+1/Mα∈ A for any α < λ, then Mis called an A-filtered module.
A continuous ascending chain (3.1) is called an A-filtration of M.
In order to determine when (S,S⊥) is a complete cotorsion theory, the fol-
lowing lemma is very effective and will be used later.
Lemma 3.1 (Eklof–Trlifaj).Let Sbe a set of modules. Then:
(1) Let Nbe an R-module. Then there exists a short exact sequence 0→
N→Q→A→0, where Q∈ S⊥and Ais an S-filtered module, and
thus A∈⊥(S⊥).
(2) (⊥(S⊥),S⊥)is a complete cotorsion theory.
Proof. See [2] or [7, Theorem 2.2]. 
In order to make Lemma 3.1 apply to the context of a class of related mod-
ules, we make corresponding modifications to it, but note that the idea belongs
to Eklof–Trlifaj essentially.
Lemma 3.2. Let S= GV(R)∗∪ S1be a set of modules, where S1⊆ FT .
(1) Let Nbe a GV-torsion-free module. Then there exists an exact sequence
(3.2) 0 →N→Q→A→0,
where Q∈ S†and Ais an S-filtered module such that A∈⊥(S†).
(2) Let Mbe an R-module. Then there exists an exact sequence
(3.3) 0 →B→P→M→0,
where P∈⊥(S†)and B∈ S†.
$KHDGRI3ULQW

WEAK w-PROJECTIVE MODULES 7
Proof. (1) Set X:= L
S∈S1
Sand Y:= L
J∈GV(R)
R/J. Then Xis a GV-torsion-
free module and Yis a GV-torsion module. Set S=X⊕Y. Then S⊥={S}⊥.
Thus we may assume that Sis the class of modules composed of the fixed
module Sand its direct sums. Let 0 →K1
µ1
−→ F1→X→0 and 0 →K2
µ2
−→
F2→Y→0 be exact sequences, where F1and F2are free modules. Set
F:= F1⊕F2and K:= K1⊕K2. Then 0 →Kµ
−→ F→S→0 is an exact
sequence, where µ:= µ1⊕µ2. Since Xis GV-torsion-free, K1is a w-module.
Since Yis GV-torsion, we have (K2)w=F2
Take a regular cardinal λso that Khas a generating system Zwith |Z|< λ.
Set Q0:= N. Then Q0is GV-torsion-free. For α < λ, if Qαhas been
constructed, select a free module F0
αand an epimorphism δα:F0
α→Qα. Set
Iα:= HomR(K, Qα) to be a new index set and define µα:K(Iα)→F(Iα)
as the homomorphism of direct sums, which is induced by µ. Then µαis a
monomorphism and Coker(µα) = S(Iα).
Define ϕα:K(Iα)⊕F0
α= ( L
f∈Iα
Kf)⊕F0
α→Qα, where Kf=K, by
ϕα([uf], z) = P
f∈Iα
f(uf) + δα(z), where uf∈Kf, z ∈F0
α. Since δαis an
epimorphism, so is ϕα. In addition, for any f∈Iα, let if:K→K(Iα)and
jf:F→F(Iα)be the natural imbeddings. Then one has
(3.4) f=ϕαifand jfµ=µαif.
Now assume that if β6α, then Qβhas been constructed (if αis a limit
ordinal, set Qα=S
β<α
Qβ), in particular, Qαhas been constructed. Construct
the following pushout diagram:
0//K(Iα)⊕F0
α
µα⊕1//
ϕα

F(Iα)⊕F0
α//
ψα

S(Iα)//
∼
=

0
0//Qα
hα//Qα+1 //Qα+1/Qα//0
One gets Qα+1. At this time ψαis an epimorphism. As you can see from the
above diagram, if Qαis a GV-torsion-free module, then Ker(ψα) = Ker(ϕα)
is a w-module, and thus Qα+1 is also a GV-torsion-free module. Hence by a
transfinite induction, we see that each Qαis a GV-torsion-free module.
Set Q:= S
α<λ
Qα= lim
→
α<λ
Qα. Then Qis a GV-torsion-free module. Set
A:= Q/N and Aα:= Qα/N . Then Aα+1/Aα∼
=Qα+1/Qα∼
=S(Iα). Since
Q=S
α<λ
Qα, one gets that A=S
α<λ
Aα. Thus Ais an S-filtered module, and
thus one has A∈⊥(S⊥). Since S†⊆ S⊥, one has A∈⊥(S†).
Let us prove that Q∈ S⊥. For this, it is sufficient to prove that µ∗:
HomR(F, Q)→HomR(K, Q) is an epimorphism. Let g:K→Qbe a ho-
momorphism. Since the generating system Zof Ksatisfies |Z|< λ and Q=
$KHDGRI3ULQW

8 H. KIM, L. QIAO, AND F. WANG
S
α<λ
Qα, there exists an ordinal α < λ such that Im(g)⊆Qα. Thus there exists
a homomorphism f:K→Qαsuch that g(x) = f(x) for any x∈K. By the
pushout diagram above and (3.4), one has ψαjfµ=ψαµαif=hαϕαif=hαf.
Define σ:F→Qby σ(z) = ψαjf(z)∈Qα+1 ⊆Q. Then one can ver-
ify directly that g=σµ =µ∗(σ). Thus µ∗is an epimorphism. Therefore
Q∈ S⊥∩ F T =S†.
(2) Take an exact sequence 0 →N→F→M→0, where Fis a projec-
tive module. Then Nis a GV-torsion-free module. By (1), there is an exact
sequence 0 →N→Q→A→0, where Q∈ S†and A∈⊥(S†). Consider the
following commutative diagram with two exact rows:
0

0

0//N//

F

//M//0
0//Q//

P//

M//0
A

A

0 0
where the square diagrams in the upper left and lower corners are pushout
diagrams. Since F , A ∈⊥(S†), one has P∈⊥(S†). Therefore one gets the
exact sequence (3.3) by taking B:= Q.
Let Abe a class of modules. Then an A-precover f:C→Mof Mis said
to be special if fis surjective and Ker(f)∈ A⊥. In other words, there is an
exact sequence 0 →K→C→M→0 with C∈ A and K∈ A⊥.
Theorem 3.3. Let S= GV(R)∗∪S1be a set of modules, where S1⊆ FT . Set
A:= ⊥(S†). If Ais closed under w-isomorphisms, then (A,A⊥)is a complete
cotorsion theory.
Proof. Note that (A,A⊥) is the cotorsion theory generated by S†. Let us prove
that any module Mhas a special A-precover.
By Lemma 3.2, there is an exact sequence (3.3), where P∈ A and B∈ S†⊆
⊥(S†)⊥=A⊥. Therefore Mhas a special A-precover. 
Proposition 3.4. Let Sbe a class of modules such that GV(R)∗⊆ S. Set
B:= ⊥(S†∞). Then:
(1) S†∞is closed under direct products, direct summands, and cokernels of
monomorphisms.
(2) Bis closed under direct sums, direct summands, kernels of epimor-
phisms, and w-isomorphisms.
(3) B†=B†∞=S†∞.
$KHDGRI3ULQW

WEAK w-PROJECTIVE MODULES 9
Proof. (1) Obviously S†∞is closed under direct products and direct summands.
Obviously S⊥∞is closed under cokernels of monomorphisms. By [12, Propo-
sition 2.2(2)], W∞is also closed under cokernels of monomorphisms. Since
S†∞=S⊥∞∩ W∞,S†∞is closed under cokernels of monomorphisms.
(2) Obviously Bis closed under direct sums and direct summands. By (1),
Bis closed under kernels of epimorphisms. By Corollary 2.5, Bis closed under
w-isomorphisms.
(3) Obviously we have that S†∞⊆⊥(S†∞)⊥∩ FT =B†. Since Bis
closed under kernels of epimorphisms, we have B⊥∞=B⊥. Thus we have
B†=B⊥∞∩ FT =B†∞. Since S ⊆ B, it follows that B†=B†∞⊆ S†∞.
Therefore B†=S†∞.
Let Mbe an R-module. Then Mis said to be w-ℵ0-generated if there exist
a countably generated free module Fand a w-epimorphism φ:F→M.
Let Mbe a w-projective w-module. If there is a continuous ascending chain
of w-projective w-submodules of M:
0 = M0⊆M0
1⊆M0
2⊆ · · · ⊆ M0
α⊆ · · · ⊆ M0
λ=M
such that each factor M0
α+1/M0
αis a w-ℵ0-generated w-projective module, then
it is said that Mhas a w-projective w-ℵ0-continuous ascending chain. It follows
from [13, Theorem 3.5] that if Mis a w-projective w-module, then Mhas a
w-projective w-ℵ0-continuous ascending chain.
Proposition 3.5. (1) wPw†=P†∞
w.
(2) Let S= GV(R)∗∪ S1be a set of modules, where S1is the class of
w-projective w-ℵ0-generated w-modules. Then S†∞=P†∞
w.
(3) Let S= GV(R)∗∪ S1be a set of modules, where S1={R}. Then
S†∞=W∞.
Proof. (1) This follows immediately from Proposition 3.4 by setting S:= Pw.
(2) Since S ⊆ Pw, we have P†∞
w⊆ S†∞. Let N∈ S †∞. For any w-
projective w-module P, by [13, Theorem 3.5] Pis an S1-filtered module. Thus
Exti
R(P, N ) = 0 for any i>1. By Proposition 2.7, Nis a strong w-module. Let
Pbe a w-projective module. Then one has the following two exact sequences:
0→torGV(R)(P)→P→P/torGV(R)(P)→0
and
0→Q→Qw→Qw/Q →0,
where Q:= P/torGV(R)(P) is GV-torsion-free. Considering two long exact se-
quences induced by the above two exact sequences, it follows that Exti
R(P, N ) =
0 for any w-projective module Pand any i>1 since Pwis closed under w-
isomorphisms. Thus N∈ P⊥∞
w∩ FT =P†∞
w. Therefore S†∞=P†∞
w.
(3) This is trivial. 
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10 H. KIM, L. QIAO, AND F. WANG
Theorem 3.6. Let S= GV(R)∗∪ S1be a set of modules, where S1⊆ FT .
Set B:= ⊥(S†∞). Then (B,B⊥)is a hereditary and complete cotorsion theory.
Proof. For any M∈ S, fix a projective resolution of M. Let LMbe the set
of all syzygies of this projective resolution of M(including Mitself as −1
syzygy). Set L:= S
M∈S
LM. Then Lis again a set. Note that Lcan be split
into L= GV(R)∗∪ L1, where L1is the set of all syzygies of M∈ S1and all
non-negative syzygies of R/J ∈GV(R)∗. Then L1⊆ F T .
Let N∈ S⊥∞. For any X∈ L, there exists an exact sequence
(3.5) 0 →X→Pk→ · · · → P1→P0→M→0,
where each Piis a projective module and M∈ S. Thus one has Ext1
R(X, N )∼
=
Extk+2
R(M, N ) = 0. Therefore N∈ L⊥.
On the other hand, let N∈ L⊥. For any Y∈ S and any k>−1, by
considering the exact sequence (3.5), one has Extk+2
R(Y, N )∼
=Ext1
R(X, N ) = 0.
Thus N∈ S⊥∞. Therefore L⊥=S⊥∞. By Theorem 3.3, (B,B⊥) is a complete
cotorsion theory. It follows by Proposition 3.4 that (B,B⊥) is a hereditary
cotorsion theory. 
Now we are ready to state the main theorem.
Theorem 3.7. (wPw,wPw⊥)is a hereditary and complete cotorsion theory,
and so every module has a special weak w-projective precover.
Proof. Let S1be the collection of all w-ℵ0-generated w-projective w-modules
and set S:= GV(R)∗∪ S1. Since the collection of all ℵ0-generated modules
is a set, Sis also a set. By Proposition 3.5(2), S†∞=W∞. By Theorem 3.6,
(wPw,wPw⊥) is a hereditary and complete cotorsion theory. 
According to [5, 6], we say that a module Mis a w∞-projective module if
Ext1
R(M, N ) = 0 for any strong w-module N. Denote by Pw∞the class of
w∞-projective modules. Then Pw∞=⊥W∞.
Theorem 3.8. (Pw∞,P⊥
w∞)is a hereditary and complete cotorsion theory.
Proof. Set S1:= {R}and S:= GV(R)∗∪ S1. Then Sis a set of modules. By
Proposition 3.5, S†∞=W∞. Thus Pw∞=⊥(S†∞). Now the assertion follows
by Theorem 3.6. 
Proposition 3.9. Let Mbe a w-module. Then there is a special weak w-
projective precover of M,ϕ:P→Msuch that Pis a w-module and Ker(ϕ)∈
P†∞
w.
Proof. We use the notation Las in the proof of Theorem 3.6 and the notation
Sas in Proposition 3.5(2). Then L†=S†∞=P†∞
w. Now the assertion follows
by Theorem 3.6. 
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WEAK w-PROJECTIVE MODULES 11
Recall that a class of modules is said to be hereditary if it is closed under
isomorphic copies and submodules.
Lemma 3.10. If Pwis a hereditary class of modules, then wPw†=Pw†.
Proof. If Pwis a hereditary class of modules, then P⊥
w=P⊥∞
w, and thus
P†
w=P†∞
w. Now the assertion immediately follows by applying Proposition
3.5(1). 
In the following result, we give some necessary and sufficient conditions for
weak w-projective modules to be w-projective.
Theorem 3.11. The following conditions are equivalent for a ring R:
(1) Every weak w-projective module is w-projective.
(2) Every weak w-projective w-module is w-projective.
(3) (Pw,Pw⊥)is a hereditary cotorsion theory and every w-module has a
special Pw-precover of a w-module.
Proof. (1)⇒(3) This follows by Theorem 3.7 and Proposition 3.9.
(3)⇒(2) Let Mbe a weak w-projective w-module. By assumption, there
is an exact sequence 0 →A→P→M→0 such that Pis a w-projective
w-module and A∈ Pw⊥. Since any GV-torsion module is w-projective, Ais a
w-module. By Lemma 3.10, A∈ Pw†=Pw†∞. Thus Ext1
R(M, A) = 0, and so
the above exact sequence is split. Therefore Mis a w-projective module.
(2)⇒(1) Let Mbe a weak w-projective module. It follows from [12, Corollary
2.7] that L(M) is a weak w-projective module. By assumption, L(M) is a w-
projective module. So Mis a w-projective module. 
Proposition 3.12. Let Abe a class of modules which is closed under w-
isomorphisms. Let Mbe a GV-torsion-free module and ϕ:P→Mbe an
A-cover. Then:
(1) Pis a GV-torsion-free module.
(2) If ϕis a special A-cover and Mis a w-module, then Pis a w-module.
Proof. Set T:= torGV (P) and B:= P /T . Then Bis a GV-torsion-free module.
Let π:P→Bbe a natural homomorphism. Since Mis a GV-torsion-free
module, ϕinduces a homomorphism ψ:B→Msuch that ψ(x) = ϕ(x) for
any x∈F, that is ψπ =ϕ. Since Ais closed under w-isomorphisms, it follows
that B∈ A. Thus there is a homomorphism h:B→Psuch that ϕh =ψ. So
ϕhπ =ψπ =ϕ. Hence hπ is an isomorphism, and thus πis an isomorphism.
Therefore Pis a GV-torsion-free module.
(2) By (1), A:= Ker(ϕ) is also a GV-torsion-free module. Since Ais closed
under w-isomorphisms, Acontains all GV-torsion modules. So Ais a w-module.
It follows from the exact sequence 0 →A→P→M→0 that Pis a w-
module. 
Theorem 3.13. Let Abe a class of modules closed under w-isomorphisms.
Let Mbe a GV-torsion-free module. Then Mhas a special A-cover if and only
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12 H. KIM, L. QIAO, AND F. WANG
if Mwhas a special A-cover. In addition, if Mis GV-torsion-free and Bis a
special A-cover of M, then Bwis a special A-cover of Mw.
Proof. Let ϕ:P→Mwbe an A-cover of Mw. Set T:= Mw/M. Then
Tis a GV-torsion module. Let π:Mw→Tbe a natural homomorphism.
Set g:= πϕ,A:= Ker(ϕ), and B:= Ker(g). Then one has the following
commutative diagram with exact rows and columns:
0

0

A

A

0//B//
ϕ0
Pg//
ϕ

T//0
0//M//

Mw
π//

T//0
0 0
where ϕ0=ϕ|B. It follows that ϕ0:B→Mis a special A-precover of M.
Let h:B→Bbe a homomorphism such that ϕ0h=ϕ0. By [10, Theorem
6.3.2], hcan be extended only to a homomorphism h0:P→P. So ϕh0is
an extension of ϕ0h. Again by [10, Theorem 6.3.2], ϕh0=ϕ. So h0is an
isomorphism. Thus his a monomorphism.
Let x∈B. Then there is y∈Psuch that h0(y) = x. So gh0(y) = πϕh0(y) =
πϕ(y) = g(y). Therefore b:= y−h0(y) = y−x∈Ker(g) = B. So y=b+x∈B,
which results in x=h(y). Thus his an epimorphism. So his an isomorphism,
and thus ϕ0:B→Mis an A-cover of M.
Conversely, let α:B→Mbe an A-cover of Mand P:= Bw. It follows
from Proposition 3.12(1) that Bis a GV-torsion-free module. By [10, Theorem
6.3.2], αinduces a unique homomorphism ϕ:P→Mw. Set T:= P /B and
T2:= Mw/M. Then Tand T2are GV-torsion modules. Thus one has the
following commutative diagram with two exact rows:
0//B//
α

Pπ//
ϕ

T//
β

0
0//M//Mw
π1//T2//0
Set A:= Ker(α), D:= Ker(ϕ), and T1:= Ker(β). It follows from the snake
lemma that one has the following exact sequence: 0 →A→D→T1→0.
Because A∈ A⊥, one has Ext1
R(T1, A) = 0. Thus D∼
=A⊕T1. Since Dis GV-
torsion-free, it follows that T1= 0, and so D=A. Since αis an epimorphism,
ϕis also an epimorphism, and thus βis an isomorphism. Hence ϕis a special
A-precover of Mw.
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WEAK w-PROJECTIVE MODULES 13
Now let h:P→Pbe a homomorphism such that ϕh =ϕ. Consider the
following diagram with exact two rows:
0//B//
h0
Pπ//
h

T//0
0//B//Pπ//T//0
Then πh =β−1π1ϕh =β−1π1ϕ=π, and so the square diagram on the right is a
commutative diagram. Thus h0:B→Bmakes the left square a commutative
diagram. Since αis the restriction of ϕon B, one has αh0=α. So h0is
an isomorphism, and thus his an isomorphism. Therefore ϕis an A-cover of
Mw.
Proposition 3.14. Let Abe a class of modules which is closed under w-
isomorphisms. Let Mbe an R-module and set T:= torGV(M). If ϕ:P→
M/T is a special A-cover which makes the pul lback diagram:
0//Tλ//P1
β//
α

P//
ϕ

0
0//T//Mπ//M/T //0,
then α:P1→Mis a special A-cover.
Proof. Because P1is w-isomorphic to P, one has P1∈ A. Set A:= Ker(ϕ).
Since Ker(α)∼
=A, it follows that α:P1→Mis a special A-precover. Let
h:P1→P1be a homomorphism such that αh =α. It follows from Proposition
3.12(1) that Pis a GV-torsion-free module. Thus hinduces a homomorphism
h:P→Psuch that ϕh =ϕ. So his an isomorphism. Thus one has the
following commutative diagram with two exact rows:
0//T//P1//
h

P//
h

0
0//T//P1//P//0
So his an isomorphism. Therefore αis a special A-cover. 
Remark 3.15.Taking A:= wPw, by Theorem 3.13 and Proposition 3.14, in
order to discuss the existence of a weak w-projective cover of a module, just
consider whether the w-module has a weak w-projective cover.
Acknowledgements. The authors would like to express their sincere thanks
for the referee for his/her careful reading and helpful comments. This research
was supported by the Academic Research Fund of Hoseo University in 2019
(20190817).
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14 H. KIM, L. QIAO, AND F. WANG
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Hwankoo Kim
Division of Computer Engineering
Hoseo University
Asan 31499, Korea
Email address:hkkim@hoseo.edu
Lei Qiao
College of Mathematics and Software Science
Sichuan Normal University
Chengdu 610068, P. R. China
Email address:lqiao@sicnu.edu.cn
Fanggui Wang
College of Mathematics and Software Science
Sichuan Normal University
Chengdu 610068, P. R. China
Email address:wangfg2004@163.com
$KHDGRI3ULQW