ALMOST F-INJECTIVE MODULES AND ALMOST FLAT MODULES Zhu Zhanmin 1. Introduction ...

12 Pages·2014·0.11 MB·English

InternationalElectronicJournalofAlgebra
Volume16(2014)115-126
ALMOST F-INJECTIVE MODULES AND ALMOST FLAT MODULES
ZhuZhanmin
Received:16May2014;Revised:28June2014
CommunicatedbyAbdullahHarmancı
Abstract. A left R-module M is called almost F-injective, if every R-homomorphism
fromafinitelypresentedleftidealtoMextendstoahomomorphismofRtoM. Aright
R-moduleVissaidtobealmostflat,ifforeveryfinitelypresentedleftidealI,thecanonical
mapV⊗I→V⊗Rismonic.AringRiscalledleftalmostsemihereditary,ifeveryfinitely
presentedleftidealofRisprojective. AringRissaidtobeleftalmostregular,ifevery
finitelypresentedleftidealofRisadirectsummandofRR. Weobservesomecharacter-
izationsandpropertiesofalmostF-injectivemodulesandalmostflatmodules. Usingthe
conceptsofalmostF-injectivityandalmostflatnessofmodules,wepresentsomecharac-
terizationsofleftcoherentrings,leftalmostsemihereditaryrings,andleftalmostregular
rings.
MathematicsSubjectClassification(2010):16E50,16E60,16D40,16D50,16P70
Keywords:AlmostF-injectivemodules,almostflatmodules,coherentrings,almostsemi-
hereditaryrings,almostregularrings
1. Introduction
Throughoutthispaper,Rdenotesanassociativeringwithidentityandallmodulescon-
sideredareunitary. ForanyR-module M, M+ =Hom(M,Q/Z)willbethecharactermod-
uleofM.
Recall that a left R-module A is said to be finitely presented if there is an exact se-
quence F → F → A → 0 in which F ,F are finitely generated free left R-modules,
1 0 1 0
or equivalently, if there is an exact sequence P → P → A → 0, where P ,P are
1 0 1 0
finitely generated projective left R-modules. Let n be a positive integer. Then a left
R-module M is called n-presented [2] if there is an exact sequence of left R-modules
F → F → ··· → F → F → M → 0 in which every F is a finitely generated
n n−1 1 0 i
free,equivalently,projectiveleftR-module. AleftR-module M issaidtobeFP-injective
[9]ifExt1(A,M) = 0foreveryfinitelypresentedleftR-module A. FP-injectivemodules
arealsocalledabsolutelypuremodules[7]. AleftR-module M iscalledF-injective[4]if
every R-homomorphism from a finitely generated left ideal to M extends to a homomor-
phismofRtoM,orequivalently,ifExt1(R/I,M)=0foreveryfinitelygeneratedleftideal
I. ArightR-moduleMisflatifandonlyifTor (M,A)=0foreveryfinitelypresentedleft
1
116 ZHUZHANMIN
R-module A if and only if Tor (M,R/I) = 0 for every finitely generated left ideal I. We
1
recallalsothataringRiscalledleftcoherent ifeveryfinitelygeneratedleftidealofRis
finitelypresented.Coherentringsandtheirgenerationshavebeenstudiedbymanyauthors
(see,forexample,[1,2,6,7,9,10]).
Inthispaper, weshallextendtheconceptsof F-injectivemodulesandflatmodulesto
almost F-injectivemodulesandalmostflatmodulesrespectively, andweshallgivesome
characterizations and properties of almost F-injective modules and almost flat modules.
Moreover,weshallcallaringRleftalmostsemihereditaryifeveryfinitelypresentedleft
ideal of R is projective. And we shall call a ring R left almost regular if every finitely
presentedleftidealofRisadirectsummand. Leftcoherentrings,leftalmostsemihered-
itary rings and left almost regular rings will be characterized by almost F-injective left
R-modulesandalmostflatrightR-modules.
2. AlmostF-injectivemodulesandalmostflatmodules
WefirstextendtheconceptofF-injectivemodulesasfollows.
Definition2.1. AleftR-moduleMiscalledalmostF-injective,ifeveryR-homomorphism
fromafinitelypresentedleftidealtoMextendstoahomomorphismofRtoM.
Recall that a submodule A of a left R-module B is said to be a pure submodule if for
everyrightR-module M, theinducedmap M⊗ A → M⊗ Bismonic, orequivalently,
R R
everyfinitelypresentedleftR-moduleisprojectivewithrespecttotheexactsequence0→
A → B → B/A → 0. In this case, the exact sequence 0 → A → B → B/A → 0 is
called pure. It is well known that a left R-module M is FP-injective if and only if it is
pureineverymodulecontainingitasasubmodule. Wecallashortexactsequenceofleft
R-modules 0 → A → B → C → 0 almost pure if for every finitely presented left ideal
I, R/I is projective with respect to this sequence. In this case, we call A an almost pure
submoduleofB.
Next,wegivesomecharacterizationsofalmostF-injectivemodules.
Theorem2.2. LetMbealeftR-module. Thenthefollowingstatementsareequivalent.
(1) MisalmostF-injective.
(2) Ext1(R/I,M)=0foreveryfinitelypresentedleftidealI.
(3) M isinjectivewithrespecttoeveryexactsequence0 → C → B → R/I → 0of
leftR-moduleswithIafinitelypresentedleftideal.
(4) M isinjectivewithrespecttoeveryexactsequence0 → K → P → R/I → 0of
leftR-moduleswithPprojectiveandIafinitelypresentedleftideal.
(5) EveryexactsequenceofleftR-modules0→ M → M(cid:48) → M(cid:48)(cid:48) →0isalmostpure.
ALMOSTF-INJECTIVEMODULESANDALMOSTFLATMODULES 117
(6) MisanalmostpuresubmoduleofitsinjectiveenvelopeE(M).
(7) There exists an almost pure exact sequence 0 → M → M(cid:48) → M(cid:48)(cid:48) → 0 of left
R-moduleswithM(cid:48)injective.
(8) There exists an almost pure exact sequence 0 → M → M(cid:48) → M(cid:48)(cid:48) → 0 of left
R-moduleswithM(cid:48)almostF-injective.
Proof. (1)⇔(2)BytheexactsequenceHom(R,M)→Hom(I,M)→Ext1(R/I,M)→0.
(2)⇒(3)BytheexactsequenceHom(B,M)→Hom(C,M)→Ext1(R/I,M)=0.
(3)⇒(4)⇒(1)areclear.
(2)⇒(5)Assume(2). Thenwehaveanexactsequence
Hom(R/I,M(cid:48))→Hom(R/I,M(cid:48)(cid:48))→Ext1(R/I,M)=0
foreveryfinitelypresentedleftidealI,andso(5)follows.
(5)⇒(6)⇒(7)⇒(8)areobvious.
f
(8) ⇒ (2)By(8),wehaveanalmostpureexactsequence0 → M → M(cid:48) → M(cid:48)(cid:48) → 0
ofleftR-moduleswhere M(cid:48) isalmost F-injective,andso,foreveryfinitelypresentedleft
ideal I,wehaveanexactsequenceHom(R/I,M(cid:48)) →f∗ Hom(R/I,M(cid:48)(cid:48)) → Ext1(R/I,M) →
Ext1(R/I,M(cid:48))=0with f epic. ThisimpliesthatExt1(R/I,M)=0,and(2)follows. (cid:3)
∗
Proposition 2.3. The class of almost F-injective left R-modules is closed under direct
limitsandalmostpuresubmodules.
Proof. ItiseasytoseethattheclassofalmostF-injectiveleftR-modulesisclosedunder
direct limits by [1, Lemma 2.9(2)] and Theorem 2.2(2). Now let A be an almost pure
submoduleofanalmost F-injectiveleftR-module B. Foranyfinitelypresentedleftideal
I,wehaveanexactsequence
Hom(R/I,B)→Hom(R/I,B/A)→Ext1(R/I,A)→Ext1(R/I,B)=0.
SinceAisalmostpureinB,thesequenceHom(R/I,B)→Hom(R/I,B/A)→0isexact.
HenceExt1(R/I,A)=0,andsoAisalmostF-injective. (cid:3)
Proposition 2.4. Let {M | i ∈ I} be a family of left R-modules. Then the following
i
statementsareequivalent.
(1) EachM isalmostF-injective.
i
(cid:81)
(2) M isalmostF-injective.
i∈I i
(3) ⊕ M isalmostF-injective.
i∈I i
Proof. (1)⇔(2)bytheisomorphismExt1(A,(cid:81) M) (cid:27) (cid:81) Ext1(A,M)andTheorem
i∈I i i∈I i
2.2(2). (1)⇔(3)isobvious. (cid:3)
118 ZHUZHANMIN
Definition2.5. ArightR-moduleV issaidtobealmostflat,ifforeveryfinitelypresented
leftidealI,thecanonicalmapV⊗I →V⊗Rismonic.
Clearly, a right R-module V is almost flat if and only if Tor (V,R/I) = 0 for every
1
finitelypresentedleftidealI.
Proposition 2.6. Let {V | i ∈ I} be a family of right R-modules. Then the following
i
statementsareequivalent.
(1) EachV isalmostflat.
i
(2) ⊕ V isalmostflat.
i∈I i
(cid:81)
(3) V isalmostflat.
i∈I i
Proof. (1)⇔(2)bytheisomorphismTor (⊕ V,A)(cid:27)⊕ Tor (V,A).
1 i∈I i i∈I 1 i
(1) ⇔ (3) For any finitely presented right ideal T, by [1, Lemma 2.10], there is an
isomorphism Tor ((cid:81) V,R/T) (cid:27) (cid:81) Tor (V,R/T), so the conditions (1) and (3) are
1 i∈I i i∈I 1 i
equivalent. (cid:3)
Theorem2.7. ThefollowingaretrueforanyringR.
(1) AleftR-moduleMisalmostF-injectiveifandonlyifM+isalmostflat.
(2) ArightR-moduleMisalmostflatifandonlyifM+isalmostF-injective.
(3) The class of almost F-injective left R-modules is closed under pure submodules,
purequotients,directsums,directsummands,directproductsanddirectlimits.
(4) The class of almost flat right R-modules is closed under pure submodules, pure
quotients,directsums,directsummands,directproductsanddirectlimits.
Proof. (1) Let I be a finitely presented left ideal. Then R/I is 2-presented. So, by [1,
Lemma2.7(2)],wehave
Tor (M+,R/I)(cid:27)Ext1(R/I,M)+,
1
andthen(1)follows.
(2)Thisfollowsfromthedualityformula
Extn(N,M+)(cid:27)Tor (M,N)+,
n
whereMisarightR-moduleandN isaleftR-module.
(3) By Proposition 2.3 and Proposition 2.4, we need only to prove that the class of
almostF-injectiveleftR-modulesisclosedunderpurequotients. Let0→ A→ B→C →
0beapureexactsequenceofleftR-moduleswith Balmost F-injective. Thenwegetthe
splitexactsequence0 →C+ → B+ → A+ → 0. Since B+ isalmostflatby(1),C+ isalso
almostflat,andsoCisalmostF-injectiveby(1)again.
ALMOSTF-INJECTIVEMODULESANDALMOSTFLATMODULES 119
(4) By Proposition 2.6, the class of almost flat right R-modules is closed under direct
sums, directsummandsanddirectproducts. Let0 → A → B → C → 0beapureexact
sequenceofrightR-moduleswithBalmostflat. SinceB+ isalmostF-injectiveby(2),A+
andC+ are also almost F-injective, and so A andC are almost flat by (2) again. So the
classofalmostflatrightR-modulesisclosedunderpuresubmodulesandpurequotients.
Bytheisomorphismformula
Tor (N,limM )(cid:27)limTor (N,M ),
n −−→ k −−→ n k
weseethattheclassofalmostflatrightR-modulesisclosedunderdirectlimits. (cid:3)
Theorem2.8. ThefollowingstatementsareequivalentforaringR.
(1) Risleftcoherent.
(2) EveryalmostF-injectiveleftR-moduleisF-injective.
(3) EveryalmostF-injectiveleftR-moduleisFP-injective.
(4) EveryalmostflatrightR-moduleisflat.
Proof. (1)⇒(2)and(3)⇒(2)areobvious.
(2) ⇒ (4) Let M be an almost flat right R-module. Then by Theorem 2.7(2), M+ is
almostF-injective,soM+isF-injectiveby(2). AndthusMisflat.
(4) ⇒ (1)Assume(4). ThensincethedirectproductsofalmostflatrightR-modulesis
almostflat,thedirectproductsofflatrightR-modulesisflat,andsoRisleftcoherent.
(1),(2) ⇒ (3)Bytheproofof[7, Theorem4], every F-injectiveleftR-moduleovera
leftcoherentringRisFP-injective,andso(3)followsfrom(1)and(2). (cid:3)
LetF beaclassofleft(right)R-modulesandMaleft(right)R-module. Following[3],
wesaythatahomomorphismϕ : M → F where F ∈ F isanF-preenvelopeof M iffor
any morphism f : M → F(cid:48) with F(cid:48) ∈ F, there is a g : F → F(cid:48) such that gϕ = f. An
F-preenvelopeϕ: M → F issaidtobeanF-envelopeifeveryendomorphismg: F → F
suchthatgϕ = ϕisanisomorphism. Dually,wehavethedefinitionsofF-precoversand
F-covers. F-envelopes (F-covers) may not exist in general, but if they exist, they are
uniqueuptoisomorphism.
Theorem2.9. ThefollowingholdforanyringR.
(1) Every left R-module has an almost F-injective cover and an almost F-injective
preenvelope.
(2) EveryrightR-modulehasanalmostflatcoverandanalmostflatpreenvelope.
(3) If A → Bis an almost F-injective (resp. almost flat) preenvelope of a left (resp.
right) R-module A, then B+ → A+ is an almost flat (resp. almost F-injective)
precoverofA+.
120 ZHUZHANMIN
Proof. (1)SincetheclassofalmostF-injectiveleftR-modulesisclosedunderdirectsums
andpurequotientsbyTheorem2.7(3),everyleftR-modulehasanalmostF-injectivecover
by[5,Theorem2.5]. Sincetheclassofalmost F-injectiveleftR-modulesisclosedunder
direct summands, direct products and pure submodules by Theorem 2.7(3), every left R-
modulehasanalmostF-injectivepreenvelopeby[8,Corollary3.5(c)].
(2)issimilartothatof(1).
(3) Let A → B be an almost F-injective preenvelope of a left R-module A. Then B+
isalmostflatbyTheorem2.7(1). ForanyalmostflatrightR-module M, M+ isanalmost
F-injectiveleftR-modulebyTheorem2.7(2),andsoHom(B,M+)→Hom(A,M+)isepic.
Considerthefollowingcommutativediagram:
Hom(B,M+) −−−−−−→ Hom(A,M+)
τ1(cid:121) (cid:121)τ2
Hom(M,B+) −−−−−−→ Hom(M,A+)
Sinceτ andτ areisomorphisms,Hom(M,B+)→Hom(M,A+)isanepimorphism. So
1 2
B+ → A+isanalmostflatprecoverofA+. Theotherissimilar. (cid:3)
Proposition2.10. ThefollowingstatementsareequivalentforaringR.
(1) RisalmostF-injective.
R
(2) EveryleftR-modulehasanepicalmostF-injectivecover.
(3) EveryrightR-modulehasamonicalmostflatpreenvelope.
(4) EveryinjectiverightR-moduleisalmostflat.
(5) EveryFP-injectiverightR-moduleisalmostflat.
Proof. (1)⇒(2)Let M bealeftR-module. Then M hasanalmost F-injectivecoverϕ :
α
C → MbyTheorem2.9. Ontheotherhand,thereisanexactsequenceA→ M →0with
Afree. NotethatAisalmostF-injectiveby(1),thereexistsahomomorphismβ : A →C
suchthatα=ϕβ. Thisfollowsthatϕisepic.
(2)⇒(1)Let f : N → RbeanepicalmostF-injectivecover. Thentheprojectivityof
R
Rimpliesthat RisisomorphictoadirectsummandofN,andso RisalmostF-injective.
R R R
(1) ⇒ (3) Let M be any right R-module. Then M has an almost flat preenvelope f :
M → F byTheorem2.9(2). Since( R)+ isacogenerator, thereexistsanexactsequence
R
0 → M →g (cid:81)( R)+. Since RisalmostF-injective,byTheorem2.7(1)andTheorem2.6,
R R
(cid:81)( R)+ isalmostflat,andsothereexistsarightR-homomorphismh: F →(cid:81)( R)+ such
R R
thatg=hf,whichshowsthat f ismonic.
(3)⇒(4)Assume(3).ThenforeveryinjectiverightR-moduleE,Ehasamonicalmost
flatpreenvelopeF,soEisisomorphictoadirectsummandofF,andthusEisalmostflat.
ALMOSTF-INJECTIVEMODULESANDALMOSTFLATMODULES 121
(4)⇒(1)Since( R)+isinjective,by(4),itisalmostflat. Thus RisalmostF-injective
R R
byTheorem2.7(1).
(4) ⇒ (5) Let M be an FP-injective right R-module. Then M is a pure submodule of
itsinjectiveenvelope E(M). By(4), E(M)isalmostflat. So M isalmostflatbyTheorem
2.7(4).
(5)⇒(4)isclear. (cid:3)
Theorem2.11. Let F beanalmostflatmoduleand0 → K → F → B → 0beanexact
sequenceofrightR-modules. Thenthefollowingconditionsareequivalent.
(1) Bisalmostflat.
(2) ForeveryfinitelypresentedleftidealI,thecanonicalmapK⊗R/I → F⊗R/I is
amonomorphism.
(3) ForeveryfinitelypresentedleftidealI =Rc +Rc +···+Rc ,K∩FnC = KnC,
1 2 n
whereC =(c ,c ,··· ,c )(cid:48).
1 2 n
(4) K∩FI = KIforeveryfinitelypresentedleftidealI.
Proof. (1)⇔(2)Thisfollowsfromtheexactsequence
0=Tor (F,R/I)→Tor (B,R/I)→ K⊗R/I → F⊗R/I.
1 1
(2)⇒(3)Letk∈ K∩FnC. Thenthereexistx ,x ,··· ,x ∈ F suchthatk=(cid:80)n x c ,
1 2 n j=1 j j
andsowehavek⊗(1+I) = (cid:80)n (x ⊗(c +I)) = (cid:80)n (x ⊗0) = 0in F ⊗R/I. By(2),
j=1 j j j=1 j
k⊗(1+I) = 0in K ⊗R/I. Thisimpliesthatk ∈ KI ⊆ KnC sincethemapϕ : K/KI →
K⊗R/I definedby x+KI (cid:55)→ x⊗(1+I)isanisomorphism. ThusK∩FnC ⊆ KnC. But
KnC ⊆ K∩FnC,soK∩FnC = KnC.
(3)⇒(4)LetI =Rc +Rc +···+Rc beafinitelypresentedleftideal. Letk∈ K∩FI.
1 2 n
Then there exist x ,x ,··· ,x ∈ F such that k = (cid:80)n x c . WriteC = (c ,c ,··· ,c )(cid:48).
1 2 n j=1 j j 1 2 n
Thenk∈ K∩FnC,andsok∈ KnCby(3). Thus,k∈ KI. Theinverseinclusionisclear.
(4)⇒(2)Let I = Rc +Rc +···+Rc beafinitelypresentedleftideal. If(cid:80)s k ⊗
1 2 n i=1 i
(r +I)=0inF⊗R/I,wherek ∈ K, r ∈R,then((cid:80)s kr)⊗(1+I)=0inF⊗R/I. So
i i i i=1 i i
(cid:80)s kr ∈ K∩FI. By(4),(cid:80)s kr ∈ KI,andsothereexistu ,u ,··· ,u ∈ K suchthat
i=1 i i i=1 i i 1 2 n
(cid:80)s kr = (cid:80)n uc. Thus(cid:80)s k ⊗(r +I) = (cid:80)n u ⊗(c +I) = 0in K ⊗R/I, and(2)
i=1 i i i=1 i i i=1 i i i=1 i i
follows. (cid:3)
3. Almostsemihereditaryringsandalmostregularrings
RecallthataringRiscalledleftsemihereditary[4]ifeveryfinitelygeneratedleftideal
ofRisprojective,orequivalently,ifeveryfinitelygeneratedsubmoduleofaprojectiveleft
R-modulesisprojective. Next,wedefineleftalmostsemihereditaryringsasfollows.
122 ZHUZHANMIN
Definition3.1. AringRiscalledleftalmostsemihereditary,ifeveryfinitelypresentedleft
idealofRisprojective.
Clearly,aringRisleftsemihereditaryifandonlyifRisleftalmostsemihereditaryand
leftcoherent.
Theorem3.2. ThefollowingstatementsareequivalentforaringR.
(1) Risleftalmostsemihereditary.
(2) pd (R/I)≤1foreveryfinitelypresentedleftidealIofR.
R
(3) EverysubmoduleofanalmostflatrightR-modulesisalmostflat.
(4) EveryfinitelygeneratedrightidealofRisalmostflat.
(5) EveryquotientmoduleofanalmostF-injectiveleftR-moduleisalmostF-injective.
(6) EveryquotientmoduleofanF-injectiveleftR-moduleisalmostF-injective.
(7) EveryquotientmoduleofaninjectiveleftR-moduleisalmostF-injective.
(8) EveryleftR-modulehasamonicalmostF-injectivecover.
(9) EveryrightR-modulehasanepicalmostflatenvelope.
(10) For every left R-module A, the sum of an arbitrary family of almost F-injective
submodulesofAisalmostF-injective.
Proof. (1) ⇔ (2) It follows from the exact sequence 0 = Ext1(R,M) → Ext1(I,M) →
Ext2(R/I,M)→Ext2(R,M)=0.
(1)⇒(3)LetAbeasubmoduleofanalmostflatrightR-moduleB.Thenforanyfinitely
presentedleftidealI,by(1),Iisprojectiveandhenceflat.Sotheexactnessofthesequence
0 = Tor (B/A,R) → Tor (B/A,R/I) → Tor (B/A,I) = 0impliesthatTor (B/A,R/I) = 0.
2 2 1 2
And thus from the exactness of the sequence 0 = Tor (B/A,R/I) → Tor (A,R/I) →
2 1
Tor (B,R/I)=0wehaveTor (A,R/I)=0,thisfollowsthatAisalmostflat.
1 1
(3)⇒(4)and(5)⇒(6)⇒(7)aretrivial.
(4)⇒(1)Let I beafinitelypresentedleftideal. Thenforanyfinitelygeneratedright
ideal K, the exact sequence 0 → K → R → R/K → 0 implies the exact sequence
0 → Tor (R/K,R/I) → Tor (K,R/I) = 0since K isalmostflat. SoTor (R/K,R/I) = 0,
2 1 2
and then we obtain an exact sequence 0 = Tor (R/K,R/I) → Tor (R/K,I) → 0. Thus,
2 1
Tor (R/K,I)=0,andsoI isafinitelypresentedflatleftR-module. Therefore,I isprojec-
1
tive.
(2)⇒(5)LetMbeanalmostF-injectiveleftR-moduleandNasubmoduleofM. Then
for any finitely presented left ideal I, by (2), Ext2(R/I,N) = 0. Thus the exact sequence
0 = Ext1(R/I,M) → Ext1(R/I,M/N) → Ext2(R/I,N) = 0impliesthatExt1(R/I,M/N) =
0. Consequently,M/N isalmostF-injective.
ALMOSTF-INJECTIVEMODULESANDALMOSTFLATMODULES 123
(7) ⇒ (1) Let I be a finitely presented left ideal. Then for any left R-module M, by
hypothesis, E(M)/M is almost F-injective, and so Ext1(R/I,E(M)/M) = 0. Thus, the
exactnessofthesequence0=Ext1(R/I,E(M)/M)→Ext2(R/I,M)→Ext2(R/I,E(M))=
0impliesthatExt2(R/I,M)=0. Andso,theexactnessofthesequence0=Ext1(R,M)→
Ext1(I,M)→Ext2(R/I,M)=0impliesthatExt1(I,M)=0,thisshowsthatIisprojective,
asrequired.
(6) ⇒ (8) Let M be a left R-module. Then by Theorem 2.9(1), there is an almost F-
injective cover f : E → M. Since im(f) is almost F-injective by (6), and f : E →
M is an almost F-injective precover, for the inclusion map i : im(f) → M, there is a
homomorphism g : im(f) → E such that i = fg. Hence f = f(gf). Observing that
f : E → M isanalmost F-injectivecoverandgf isanendomorphismof E, sogf isan
automorphismofE,andhence f : E → MisamonicalmostF-injectivecover.
(8)⇒(6)Let M beanalmost F-injectiveleftR-moduleand N beasubmoduleof M.
By(8), M/N hasamonicalmost F-injectivecover f : E → M/N. Letπ : M → M/N be
thenaturalepimorphism.Thenthereexistsahomomorphismg: M → Esuchthatπ= fg.
Thus f isanisomorphism,andwhenceM/N (cid:27) EisalmostF-injective.
(3)⇒(9)Let MbearightR-moduleandLet{K} bethefamilyofallsubmodulesof
i i∈I
M suchthat M/K isalmostflat. LetF = M/∩ K andπbethenaturalepimorphismof
i i∈I i
M to F. Defineα : F → (cid:81) M/K byα(m+∩ K) = (m+K)form ∈ M,thenαisa
i∈I i i∈I i i
(cid:81)
monomorphism. ByTheorem2.7(4), M/K isalmostflat. By(3),wehavethat F is
i∈I i
almostflat. ForanyalmostflatrightR-module F(cid:48) andanyhomomorphism f : M → F(cid:48),
since M/Ker(f) (cid:27) Im(f) ⊆ F(cid:48), M/Ker(f)isalmostflatby(3),andthusKer(f) = K for
j
some j ∈ I. Nowwedefineg : F → F(cid:48);x+∩ K (cid:55)→ f(x), thengisahomomorphism
i∈I i
suchthat f =gπ. Hence,πisanalmostflatpreenvelopeof M. Notethatepicpreenvelope
isanenvelope,soπ: M → F isanepicalmostflatenvelopeofM.
(9)⇒(3)Assume(9).ThenforanysubmoduleKofanalmostflatrightR-moduleM,K
hasanepicalmostflatenvelopeϕ: K → F. Sothereexistsahomomorphismh: F → M
such that i = hϕ, where i : K → M is the inclusion map. Thus ϕ is monic, and hence
K (cid:27) F isalmostflat.
(6)⇒(10)LetAbealeftR-moduleand{A |γ∈Γ}beanarbitraryfamilyofalmostF-
γ
injectivesubmodulesof A. Sincethedirectsumofalmost F-injectivemodulesisalmost
(cid:80) (cid:80)
F-injective and γ∈ΓAγ is a homomorphic image of ⊕γ∈ΓAγ, by (6), γ∈ΓAγ is almost
F-injective.
(10) ⇒ (7) Let E be an injective left R-module and K ≤ E. Take E = E = E,N =
1 2
E ⊕E ,D = {(x,−x) | x ∈ K}. Define f : E → N/Dby x (cid:55)→ (x ,0)+D, f : E →
1 2 1 1 1 1 2 2
N/D by x (cid:55)→ (0,x ) + D and write E = f(E),i = 1,2. Then E (cid:27) E is injective,
2 2 i i i i i
124 ZHUZHANMIN
i=1,2,andsoN/D= E +E isalmostF-injective. BytheinjectivityofE,(N/D)/E is
1 2 i i
isomorphictoadirectsummandofN/DandthusitisalmostF-injective. Nowwedefine
f : E → (N/D)/E by e (cid:55)→ f (e)+E . Then f is epic with Ker(f) = K, and therefore
1 2 1
E/K (cid:27)(N/D)/E isalmostF-injective. (cid:3)
1
Following[2],wecallaringRaleft(n,d)-ringifeveryn-presentedleftR-modulehas
projective dimension at most d. An (n,1)-domain is called an n-Pru¨fer domain [2]. By
Theorem3.2,aleft(2,1)-ringisaleftalmostsemihereditaryring.
Example3.3. LetKbeafield,andletT = K+Mbeavaluationringwithmaximalideal
M. LetkbeasubfieldofK withk(cid:44) K,andletR=k+M. Then
(1) If [K : k] = ∞, then by [2, Corollary 5.2 (1)], R is a 2-Pru¨fer domain but not a
Pru¨ferdomain. SoRisacommutativealmostsemihereditaryring,butRisnota
semihereditaryring.
(2) If[K :k]<∞andM = M2,thenby[2,Corollary5.2(1)],Risa2-Pru¨ferdomain
but not a Pru¨fer domain. So R is an almost semihereditary ring, but R is not a
semihereditaryring.
(3) If [K : k] < ∞ and M (cid:44) M2, then by [2, Corollary 5.2 (3)], R is a coherent
ring but not a 2-Pru¨fer domain. So R is a coherent ring, but R is not an almost
semihereditaryring.
Next,wegeneralizetheconceptofregularrings.
Definition3.4. AringRiscalledleftalmostregular,ifeveryfinitelypresentedleftideal
ofRisadirectsummandof R.
R
Theorem3.5. ThefollowingconditionsareequivalentforaringR.
(1) Risaleftalmostregularring.
(2) ForeveryfinitelypresentedleftidealI,R/IisaprojectiveleftR-module.
(3) EveryleftR-moduleisalmostF-injective.
(4) EveryrightR-moduleisalmostflat.
(5) Risleftalmostsemihereditaryand RisalmostF-injective.
R
(6) RisleftalmostsemihereditaryandeveryinjectiverightR-moduleisalmostflat.
Proof. (1)⇔(2)⇔(3),and(2)⇔(4)aswellas(1),(4)⇒(6)areobvious.
(5)⇔(6)byProposition2.10.
(6) ⇒ (4) Let M be any right R-module. Then for any finitely presented left ideal I,
sinceRisleftalmostsemihereditary,I isprojective,andso pd (R/I) ≤ 1. Thus,fromthe
R
exactsequence0 = Tor (R/I,E(M)/M) → Tor (R/I,M) → Tor (R/I,E(M)) = 0,weget
2 1 1
thatTor (R/I,M)=0. Therefore,Misalmostflat. (cid:3)
1