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PureMathematicsnØêÆ,2023,13(5),1355-1362
PublishedOnlineMay2023inHans.https://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2023.135138
4Œ{L
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∗
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Á‡
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…¢D{Lé"
'…c
4Œ{L§4Œ²"§{Lé
Max-CotorsionModules
JuanniYang
∗
,XiaoyanYang
CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu
Received:Apr.21
st
,2023;accepted:May22
nd
,2023;published:May29
th
,2023
Abstract
Inthispaper, westudysomecriterionsandhomologicalpropertiesofmax-cotorsion
modules.It isprovedthatthe classofmax-flat modules andthe classofmax-cotorsion
modulesisaperfectandhereditarycotorsionpair.
∗1˜Šö"
©ÙÚ^:ïV,¡ÿ.4Œ{L[J].nØêÆ,2023,13(5):1355-1362.
DOI:10.12677/pm.2023.135138
ïV§¡ÿ
Keywords
Max-CotorsionModule,Max-FlatModule,CotorsionPair
Copyright
c
2023byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense (CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
3©¥, XJvkAO•Ñ, •Ä¤k‚´(Ü‚, ¤k´j. {L´ÓN“ê†
Ø¥-‡ïÄé–ƒ˜, 3“êAÛÚ“êL«nØ¥k-‡A^, …{L3鈫‚•
x¥•åXéŒŠ^.
1959c, Harrison 3[1] ¥•• xšk•Abelian +(Ú5Ÿ, Ú\{LVg.
1996c, Xu3[2]¥XÚ?Ø{Lƒ'5Ÿ, y²²"a†{La¤{Lé,d
Xu„y²z‡R´{L…=‚R´†‚. 2000c,Trlifaj 3[3]¥y²²"C
XÚ{L•ä•35. 2005c, Mao ÚDing 3[4] ¥?˜ÚïÄ{L5Ÿ.2010c, Xiang
3[5] ¥|^²"ƒ'5Ÿ‰Ñ4Œ²"½Â,ïÄ4Œ²"5Ÿ,¿|^4
Œ²"‘ê•x4Œvà‚. 2021c, Alagoz3[6]¥|^4Œ²"Ú\4Œ{L, |^4
Œ²"Ú4Œ{L•x‚, 4Œ¢D‚, ¿y²z‡k˜‡4Œ²"CXÚ˜‡4
Œ{L•ä.
Édéu, ©?˜ÚïÄ4Œ{L, ¿a'{LéÙƒ'5Ÿ‰?Ø.•y²
4Œ²"a†4Œ{La¤…¢D{Lé.
2.ý•£
½Â2.1([7]) C´?¿a, M´R-.
(1)¡θ:C→M´MC-ýCX,XJC∈C¿…é?¿f:C
0
→M,Ù¥
C
0
∈C, •3h: C
0
→C¦θh=f.¡C-ýCXθ: C→M´MC-CX,XJ÷vθβ= θ
gÓβ´gÓ.
(2)¡ϕ:M→C´MC-ý•ä,XJC∈C¿…é?¿f:M→C
0
,Ù¥
C
0
∈C,•3g:C→C
0
¦gϕ= f.¡C-ý•äϕ:M→C´MC-•ä,XJ÷vηϕ= ϕ
gÓη´gÓ.
DOI:10.12677/pm.2023.1351381356nØêÆ
ïV§¡ÿ
½Â2.2([8]) CÚD´ü‡a.
(1)¡(C,D)•{Lé, XJC
⊥
= D,C=
⊥
D,Ù¥
C
⊥
= {M∈R-Mod|Ext
1
R
(C,M) = 0,∀C∈C},
⊥
C= {M∈R-Mod|Ext
1
R
(M,C) = 0,∀C∈C}.
(2)¡{Lé(C,D)´,XJé?¿R-M,kÜ0−→D−→C−→M−→0Ú
0 −→M−→D
0
−→C
0
−→0, Ù¥C,C
0
∈C,D,D
0
∈D.
(3)¡{Lé(C,D)´, XJz‡R-k˜‡C-CXÚ˜‡D-•ä.
(4)¡{Lé(C,D) ´¢D,XJé?¿áÜ
0 −→A−→B−→C−→0.
eB,C∈C,KA∈C.
½Â2.3([5]) ¡R-A´4Œ²", XJéR?¿4ŒnŽI,k
Tor
R
1
(A,R/I) = 0.
½Â2.4([4]) ¡R-A´{L, XJé?¿²"F, kExt
1
R
(F,A) = 0.
3.̇(J
·‚Äk‰ÑAlagoz 3[6] ¥Ú\4Œ{L½Â.
½Â3.1([6]) ¡R-B´4Œ{L, XJé?¿4Œ²"A, k
Ext
1
R
(A,B) = 0.
5P3.2d½Â•, {S}⊆{4Œ{L}⊆{{L}.
Ún3.3([9], ½n3) M´4Œ²"…=é?¿üR-B, n≥1, k
Tor
R
n
(M,B) = 0.
·K3.4éuR-B,±eA^´dµ
(1)B´4Œ{L.
DOI:10.12677/pm.2023.1351381357nØêÆ
ïV§¡ÿ
(2)éuz‡4Œ²"R-A, n≥1, kExt
n
R
(A,B) = 0.
(3)éuR-z‡Ü0 −→F−→C−→A−→0, Ù¥A´4Œ²", ¼f
Hom
R
(−,B) ±SÜ5.
y²:(1) =⇒(2)én^8B{. en= 1, Kd½ÂŒ•¤á.b(Øén−1¤á. A´
4Œ²".•ÄR-Ü
0 −→K−→A
0
−→A−→0,
Ù¥A
0
´gd.^−⊗
R
R/IŠ^þãÜ,
Tor
R
2
(A
0
,R/I) →Tor
R
2
(A,R/I) →Tor
R
1
(K,R/I) →Tor
R
1
(A
0
,R/I).
Ï•A
0
´gd,¤±kTor
R
2
(A
0
,R/I) = Tor
R
1
(A
0
,R/I) = 0. u´
Tor
R
2
(A,R/I)
∼
=
Tor
R
1
(K,R/I).
Ï•I´R?¿4ŒnŽ, ¤±R/I´ü. qÏ•A´4Œ²",¤±dÚn3.3 Œ•
Tor
R
2
(A,R/I) = 0.
u´Tor
R
1
(K,R/I) = 0. KK´4Œ²".^Hom
R
(−,B) Š^þãáÜ,
Ext
n−1
R
(A
0
,B) →Ext
n−1
R
(K,B) →Ext
n
R
(A,B) →Ext
n
R
(A
0
,B).
Ï•A
0
´gd,¤±A
0
´Ý.u´
Ext
n−1
R
(A
0
,B) =Ext
n
R
(A
0
,B) = 0.
¤±Ext
n−1
R
(K,B)
∼
=
Ext
n
R
(A,B). qÏ•K´4Œ²",¤±d8BbŒ•,
Ext
n−1
R
(K,B) = 0.
¤±Ext
n
R
(A,B) = 0.
(2)=⇒(3)•ÄR-Ü0 −→F−→C−→A−→0,Ù¥A´4Œ²".^Hom
R
(−,B)
Š^þãÜ,
0 →Hom
R
(A,B) →Hom
R
(C,B) →Hom
R
(F,B) →Ext
1
R
(A,B).
Ï•Ext
1
R
(A,B) = 0, ¤±
0 →Hom
R
(A,B) →Hom
R
(C,B) →Hom
R
(F,B) →0
DOI:10.12677/pm.2023.1351381358nØêÆ
ïV§¡ÿ
Ü.
(3)=⇒(1) A´4Œ²".•ÄR-Ü0−→F−→C−→A−→0, Ù¥C´Ý.
·‚kXeÜ
0 →Hom
R
(A,B) →Hom
R
(C,B) →Hom
R
(F,B) →Ext
1
R
(A,B) →Ext
1
R
(C,B) = 0.
dbExt
1
R
(A,B) = 0. ÏdB´4Œ{L.
·K3.5e4Œ{La'u†Úµ4, K±ed:
(1)B´4Œ{L.
(2)éz‡ÝP,P⊗
R
B´4Œ{L.
y²:(1) =⇒(2) A´4Œ²",P´˜‡ÝR-.K•3˜‡ÝP
0
¦éu,
•I8I, kR
(I)
∼
=
P⊕P
0
.Ï•R⊗
R
B
∼
=
B, ¤±
Ext
1
R
(A,B
(I)
)
∼
=
Ext
1
R
(A,R
(I)
⊗
R
B)
∼
=
Ext
1
R
(A,(P⊕P
0
)⊗
R
B)
∼
=
Ext
1
R
(A,(P⊗
R
B)⊕(P
0
⊗
R
B))
∼
=
Ext
1
R
(A,P⊗
R
B)⊕Ext
1
R
(A,P
0
⊗
R
B)
Ï•4Œ{La'u†Úµ4,¤±B
(I)
´4Œ{L.qÏ•A´4Œ²",¤±
Ext
1
R
(A,B
(I)
) = 0.ÏdExt
1
R
(A,P⊗B) = 0.P⊗
R
B´4Œ{L.
(2)=⇒(1)-P= R.Ï•R⊗
R
B
∼
=
B, ¤±B´˜‡4Œ{L.
·K3.6é?¿˜q{B
i
}
i∈I
Ù¥I´˜‡•I8,
Q
i∈I
B
i
´4Œ{L…=z‡
B
i
´4Œ{L.
y²:=⇒)A´4Œ²"R-, …
Q
i∈I
B
i
´4Œ{L.K
Ext
1
R
(A,
Y
i∈I
B
i
) = 0.
d([10], ½n2) Œ•,kg,Ó'X, Ext
1
R
(A,
Q
i∈I
B
i
)
∼
=
Q
Ext
1
R
(A,B
i
).u´
Ext
1
R
(A,B
i
) = 0.
z‡B
i
´4Œ{L.
DOI:10.12677/pm.2023.1351381359nØêÆ
ïV§¡ÿ
⇐=)A´4Œ²"R-,…z‡B
i
´4Œ{L,kExt
1
R
(A,B
i
) = 0.2dg,Ó'X
Ext
1
R
(A,
Y
i∈I
B
i
)
∼
=
Y
Ext
1
R
(A,B
i
)
ŒExt
1
R
(A,
Q
i∈I
B
i
) = 0.
Q
i∈I
B
i
´4Œ{L.
·K3.7R´˜‡‚. K±e^‡´d:
(1)z‡R-´4Œ{L.
(2)z‡4Œ²"R-´Ý.
d,eR÷vþã^‡ƒ˜,KR´˜‡‚.
y²:(1)=⇒(2)A´4Œ²"R-.d(1)Œ•,?¿R-B´4Œ{L.u´
Ext
1
R
(A,B) = 0. A´Ý.
(2) =⇒(1)A´4Œ²"R-. KA´Ý. u´é?¿R-B, k
Ext
1
R
(A,B) = 0.
KB´˜‡4Œ{LR-.
d5P3.2Œ•, B´{L, d([2],·K3.3.1) Œ•,z‡R-´{L…=R´‚.
R÷vþã^‡(1) ž,KR´˜‡‚.
Ún3.8ü«5´4Œ{L.
y²:M´4Œ²"R-, D´ü. dÓª
Ext
n
R
(M,D
+
)
∼
=
Tor
R
n
(M,D)
+
•Tor
R
n
(M,D) = 0,lExt
n
R
(M,D
+
) = 0.D
+
´4Œ{L.
½n3.9FGL«4Œ{La, FHL«4Œ²"a. K(FH,FG)´˜‡{Lé.
y²:Äky
⊥
FG= FH. M∈
⊥
FG.Ké?¿4Œ{LN, k
Ext
1
R
(M,N) = 0.
I´R?¿4ŒnŽ. KR/I´ü.dÚn3.8 Œ•,ü«5´4Œ{L. (R/I)
+
´4Œ{L.lExt
1
R
(M,(R/I)
+
) = 0.dÓª
0 = Ext
1
R
(M,(R/I)
+
)
∼
=
(Tor
R
1
(M,R/I))
+
•Tor
R
1
(M,R/I)= 0, ÏdM´4Œ²", lM∈FH. 
⊥
FG⊆FH. M´4Œ²".
DOI:10.12677/pm.2023.1351381360nØêÆ
ïV§¡ÿ
Ké?¿4Œ{LN, k
Ext
1
R
(M,N) = 0.
lM∈
⊥
FG.FH⊆
⊥
FG.nþ,
⊥
FG=FH. eyFH
⊥
= FG.M∈FH
⊥
. Ké?¿4
Œ²"N, k
Ext
1
R
(N,M) = 0.
lM∈FG.FH
⊥
⊆FG. M∈FG. Ké?¿4Œ²"N, k
Ext
1
R
(N,M) = 0.
lM∈FH
⊥
.FG⊆FH
⊥
.nþ¤ã, (FH,FG) ´˜‡{Lé.
íØ3.10(FH,FG)´˜‡{Lé.
y²:d½n3.9Œ•,(FH,FG)´˜‡{Lé.d([6],Ún2)Ú( [11],½n3.4)z‡R-
k˜‡FH-CXÚ˜‡FG-•ä. (FH,FG) ´˜‡{Lé.
íØ3.11(FH,FG)´˜‡¢D{Lé.
y²:•ÄR-Ü0−→A−→A
0
−→A
00
−→0, Ù¥A
0
,A
00
∈FH.éR?¿4ŒnŽI,
^−⊗
R
R/IŠ^þãÜ,
Tor
R
2
(A
00
,R/I) →Tor
R
1
(A,R/I) →Tor
R
1
(A
0
,R/I).
Ï•A
0
´4Œ²", ¤±Tor
R
1
(A
0
,R/I)=0.Ï•I´R?¿4ŒnŽ, ¤±R/I´ü. q
Ï•A
00
´4Œ²",dÚn3.3 Œ•
Tor
R
2
(A
00
,R/I) = 0.
Tor
R
1
(A,R/I) = 0. ÏdA´4Œ²".(FH,FG) ´˜‡¢D{Lé.
ë•©z
[1]Harrison,D.K.(1959)InfiniteAbelianGroupsandHomologicalMethods.AnnalsofMathe-
matics,69,336-391.https://doi.org/10.2307/1970188
[2]XuJ.(1996)FlatCoversofModules.In:LectureNotesinMathematics,Vol.1634,Springer-
Verlag,Berlin.
[3]Trlifaj,J.(2000)Covers,EnvelopesandCotorsionTheories.LectureNotesfortheWorkshop
“HomologicalMethodsinModuleTheory”,Cortona,10-16September2000.
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ïV§¡ÿ
[4]Mao,L.X.andDing,N.Q.(2005)NotesonCotorsionModules.CommunicationsinAlgebra,
33,349-360.https://doi.org/10.1081/AGB-200041029
[5]Xiang,Y.(2010)Max-Injective,Max-FlatModulesandMax-CoherentRing.Bulletinofthe
KoreanMathematicalSociety,47,611-622.https://doi.org/10.4134/BKMS.2010.47.3.611
[6]Alagoz,Y.andBuyukasik,E.(2021)OnMax-FatModulesandMax-CotorsionModules.Ap-
plicableAlgebrainEngineering,32,195-215.https://doi.org/10.1007/s00200-020-00482-4
[7]Enochs,E.E.andJenda,O.M.G.(2000)RelativeHomologicalAlgebra.WalterdeGruyter,
NewYork.https://doi.org/10.1515/9783110803662
[8]Enochs,E.E.(2002)Covers,EnvelopesandCotorsionTheories.NovaBiomedical,NewYork.
[9]urÚ,•Xr.&ï4Œ²"‘ê[J].Ïz“‰ÆÆ,2010,31(12):11-13.
[10]Rotman,J.J.(2009)AnIntroductiontoHomologicalAlgebra.Springer,NewYork.
https://doi.org/10.1007/b98977
[11]Holm,H.andJorgensen,P.(2008)Covers,Precovers,andPurity.IllinoisJournalofMathe-
matics,52,691-703.https://doi.org/10.1215/ijm/1248355359
DOI:10.12677/pm.2023.1351381362nØêÆ

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