On L-Injective Modules

0.25 MB·English

by  Akeel Ramadan Mehdi

L
On -Injective Modules
AkeelRamadanMehdi
DepartmentofMathematics
CollegeofEducation
Al-QadisiyahUniversity
P.O.Box88,Al-Qadisiyah,Iraq
email: [email protected]
January13,2015
5
1
Abstract
0
2
LetM ={(M,N,f,Q)|M,N,Q∈R-Mod,N≤M, f ∈Hom (N,Q)}andletL beanonempty
R
n subclassofM.JiráskointroducedtheconceptsofL-injectivemoduleasageneralizationofinjec-
a tive module as follows: a module Q is said to be L-injective if for each (B,A,f,Q)∈L, there
J
existsahomomorphismg:B→Qsuchthatg(a)= f(a), foralla∈A. Theaimofthispaperisto
1
1 studyL-injectivemodulesandsomerelatedconcepts.

] Keywordsandphrases:Injectivemodule;Baer’scriterion;GeneralizedFuchscriterion;τ-Injective
A
module;P-filter;Hereditarytorsiontheory;τ-dense;Preradicals;M-injectivemodule;Quasi-injective;
R
Naturalclass.
.
h
t 2010MathematicsSubjectClassification: Primary: 16D50;Secondary: 16D10,16S90.
a
m
[

1 Introduction
1
v
1 Injective modules can be traced to Baer [4] who studied divisible Abelian groups. Eckmann
9 andSchopf[12]introducedtheterminology"injective". AnR-moduleM issaidtobeinjectiveif,
4
foranymoduleB,everyhomomorphism f :A→M,whereAisanysubmoduleofB,extendstoa
2
homomorphismg:B→M.Manyauthorsinterestedinthisclassofmodules, forexample, Matlis
0
. [19], Faith [13] and Page and Zhou [21]. Also, many authors tried to generalize the concept of
1
injectivemodule, forexample, JohnsonandWongintroducedquasi-injectivemodulein[17]. The
0
5 notionofM-injectivemodulewasintroducedin[3].Thenotionofτ-injectivemodulewasstudiedin
1 [23]andthenotionofsoc-injectivemodulewasintroducedin[1]. Also,Zeyadain[25]introduced
: theconceptofs-injectivemodule.
v
i LetM andN bemodules. RecallthatN issaidtobeM-injectiveifeveryhomomorphismfrom
X
a submodule of M to N extends to a homomorphism from M to N [3]. A module M is said to be
r quasi-injectiveifMisM-injective.
a
Letτ =(T,F)beatorsiontheory. AsubmoduleBofamoduleAissaidtobeτ-denseinAif
A/Bisτ-torsion(i.e. A/B∈T). AsubmoduleAofamoduleBissaidtobeτ-essentialinBifitis
τ-denseandessentialinB. Atorsiontheoryτ issaidtobenoetherianifforeveryascendingchain
I ⊆I ⊆... of left ideals of R with I =(cid:83)∞ I a τ-dense left ideal in R, there exists a positive
1 2 ∞ j=1 j
integernsuchthatI isτ-denseinR.AmoduleMissaidtobeτ-injectiveifeveryhomomorphism
n
from a τ-dense submodule of B to M extends to a homomorphism from B to M, where B is any
module. Let M be an R-module. A τ-injective envelope (or τ-injective hull) of M is a τ-injective
moduleE whichisaτ-essentialextensionofM [7]. EveryR-moduleM hasaτ-injectiveenvelope
and it is unique up to isomorphism [9]. We use the notation E (M) to stand for an τ-injective
τ
envelopeofM.Aτ-injectivemoduleEissaidtobe∑-τ-injectiveifE(A)isτ-injectiveforanyindex
set A; E is said to be countably ∑-τ-injective in case E(C) is τ-injective for any countable index
setC.LetE andM bemodules. ThenE issaidtobeτ-M-injectiveifanyhomomorphismfroma
τ-densesubmoduleofM toE extendstoahomomorphismfromM toE. AmoduleE issaidtobe
τ-quasi-injectiveifE isτ-E-injective.
1
LetM ={(M,N,f,Q)|M,N,Q∈R-Mod,N≤M, f ∈Hom (N,Q)}andletL beanonempty
R
subclassofM.Jiráskoin[16]introducedtheconceptsofL-injectivemoduleasageneralization
ofinjectivemoduleasfollows: amoduleQissaidtobeL-injectiveifforeach(B,A,f,Q)∈L,
thereexistsahomomorphismg:B→Qsuchthat(g(cid:22)A)= f. Clearly,injectivemoduleandallits
generalizationsarespecialcasesofL-injectivity. TheaimofthisarticleistostudyL-injectivity
andsomerelatedconcepts.
In section two, we give some properties and characterizations of L-injective modules. For
example, in Theorem 2.12 we give a version of Baer’s criterion for L-injectivity. Also, in Theo-
rem 2.15 we extend a characterization due to [24, Theorem 2, p. 8] of L-injective modules over
commutativeNoetherianrings.
InsectionthreeweintroducetheconceptsofL-M-injectivemoduleands-L-M-injectivemod-
ule as generalizations of M-injective modules and give some results about them. For examples,
in Theorem 3.5 we prove that if L is a nonempty subclass of M satisfy conditions (α), (β) and
(γ)andM,Q∈R-ModsuchthatM satisfiescondition(EL),thenQisL-M-injectiveifandonly
if f(M)≤Q, for all f ∈HomR(EL(M),EL(Q)) with (M,L,f(cid:22)L,Q)∈L where L={m∈M |
f(m)∈Q}=M(cid:84)f−1(Q). Also, inProposition3.13wegeneralize[7, Proposition14.12, p. 66],
[6, Proposition 1, p. 1954] and Fuchs’s result in [14]. Moreover, our version of the Generalized
FuchsCriterionisgiveninProposition3.14inwhichweprovethatifL isanonemptysubclassof
M satisfyconditions(α)and(µ)andM,Q∈R-ModsuchthatM satisfiescondition(L). Thena
moduleQiss-L-M-injectiveifandonlyifforeach(R,I,f,Q)∈L withker(f)∈Ω(M), there
existsanelement x∈Qsuchthat f(a)=ax, ∀a∈I.
In section four we study direct sums of L-injective modules. In this section we prove some
results,forexample,inProposition4.3weprovethatforanyfamily{E } ofL-injectivemod-
α α∈A
ules,whereAisaninfiniteindexset,ifL satisfiesconditions(α), (µ)and(δ)and(cid:76) E is
α∈C α
anL-injectivemoduleforanycountablesubsetC ofA,then(cid:76) E isanL-injectivemodule.
α∈A α
InTheorem4.10weprovethatforanynonemptysubclassL ofM whichsatisfiesconditions(α)
and(δ)andforanynonemptyclassK ofmodulesclosedunderisomorphiccopiesandL-injective
hulls,ifthedirectsumofanyfamily{E} ofL-injectiveR-modulesinK isL-injective,then
i i∈N
everyascendingchainI1⊆I2⊆...ofleftidealsofRinHK(R)withI∞=(cid:83)∞j=1Ij s-L-denseinR,
terminates. Also, inTheorem4.12wegeneralizeresultsin[21, p. 643]and[9, Proposition5.3.5,
p. 165] in which we prove that for any nonempty subclass L of M which satisfies conditions
(α), (µ), (δ)and(I)andforanynonemptyclassK ofmodulesclosedunderisomorphiccopies
andsubmodules,ifeveryascendingchainJ ⊆J ⊆...ofleftidealsofRsuchthat(J /J)∈K ,
1 2 i+1 i
∀i∈NandJ =(cid:83)∞ J s-L-denseinRterminates,theneverydirectsumofL-injectivemodules
∞ i=1 i
inK isL-injective.
Finally, in section five we introduce the concept of ∑-L-injectivity as a generalization of ∑-
injectivity and ∑-τ-injectivity and prove Theorem 5.4 in which we generalize Faith’s result [13,
Proposition3,p. 184]and[7,Theorem16.16,p. 98].
Throughoutthisarticle,unlessotherwisespecified,Rwilldenoteanassociativeringwithnon-
zeroidentity,andallmodulesareleftunitalR-modules.Byaclassofmoduleswemeananon-empty
classofmodules. TheclassofallleftR-modulesisdenotedbyR-Modandbyℜwemeantheset
{(M,N)|N≤M,M∈R-Mod},whereN≤M isanotationmeansN isasubmoduleofM. Given
a family of modules {M} , for each j∈I,π :(cid:76) M →M denotes the canonical projection
i i∈I j i∈I i j
homomorphism.LetMbeamoduleandletY beasubsetofM.TheleftannihilatorofY inRwillbe
denotedbyl (Y), i.e.,l (Y)={r∈R|ry=0,∀y∈Y}.Givena∈M,let(Y :a)denotetheset{r∈
R R
R|ra∈Y},andletann (a):=(0:a).TherightannihilatorofasubsetIofRinMwillbedenotedby
R
r (I),i.e.,r (I)={m∈M|rm=0,∀r∈I}.Theclass{I|IisaleftidealofRsuchthatann (m)⊆
M M R
I, forsomem∈M}willbedenotedbyΩ(M).Finally,theinjectiveenvelopeofamoduleMwillbe
denotedbyE(M).
2 Some Properties and Characterizations of L-Injective Mod-
ules
LetM ={(M,N,f,Q)|M,N,Q∈R-Mod,N≤M, f ∈Hom (N,Q)}andletL beanonempty
R
subclassofM. Jiráskoin[16]introducedtheconceptofL-injectivemoduleasageneralization
2
ofinjectivemoduleasfollows: amoduleQissaidtobeL-injectiveifeverydiagram
(cid:31)(cid:127) i (cid:47)(cid:47)
N M
f
(cid:15)(cid:15)
Q
with(M,N,f,Q)∈L,canbecompletedtoacommutativediagram.
Forevery(B,A,f,Q)∈M, Jiráskoin[16]definedrL(B,A,f,Q)andsL(B,A,f,Q)asfollows:
rL(B,A,f,Q)(sL(B,A,f,Q))isthesubmoduleofBgeneratedbyalltheg(M),g∈HomR(M,B)
forwhichthereexistsacommutativediagram
(cid:31)(cid:127) i (cid:47)(cid:47)(cid:47)(cid:47)
N M
h g
(cid:15)(cid:15) (cid:31)(cid:127) i (cid:47)(cid:47)(cid:47)(cid:47) (cid:15)(cid:15)
A B
f
(cid:15)(cid:15)
Q
with(M,N,fh,Q)∈L (andN=g−1(A)).
Wewillusethefollowingabbreviations:rL(B,Q,IQ,Q)=rL(B,Q), sL(B,Q,IQ,Q)=sL(B,Q),
rL(E(Q),Q)=rL(Q)andsL(E(Q),Q)=sL(Q).
AsubmoduleN ofamoduleMissaidtobeL-denseinMifM⊆N+rL(E(M),N)[16].
Jiráskoin[16]introducedtheconceptsofL-injectiveenvelope(orL-injectivehull)ofamod-
ule M as follows: an L-injective module E is said to be an L-injective envelope of a module M
ifthereisnoproperL-injectivesubmoduleofE containingM. IfamoduleM hasanL-injective
envelope and it is unique up to isomorphic then we will use the notation EL(M) to stand for an
L-injectiveenvelopeofM.
IntheclassM wewilldefinethepartialorder(cid:22)inthefollowingway:
(M,N,f,Q)(cid:22)(M(cid:48),N(cid:48),f(cid:48),Q(cid:48))⇐⇒M=M(cid:48), N⊆N(cid:48), Q=Q(cid:48), f(cid:48)(cid:22)N= f.
ThefollowingconditionsonL willbeusefullater,whereL alwaysdenotesanonemptysub-
classofM.
(α)(M,N,f,Q)∈L,(M,N(cid:48),f(cid:48),Q)∈M and(M,N,f,Q)(cid:22)(M,N(cid:48),f(cid:48),Q)implies(M,N(cid:48),f(cid:48),Q)∈
L,
(β) (M,N,f,A)∈L,i:A→B implies (M,N,if,B)∈L, where i is an inclusion homomor-
phism,
(γ)(M,N,f,A)∈L,g:A→Banisomorphism,implies(M,N,gf,B)∈L,
(δ)(M,N,f,A)∈L,g:A→Bahomomorphism,implies(M,N,gf,B)∈L,
(λ)(M,N,f,A)∈L,g:A→Basplitepimorphism,implies(M,N,gf,B)∈L,
(µ)(M,N,f,Q)∈L,implies(R,(N :x),f ,Q)∈L,∀x∈M,where f :(N :x)→Qisaho-
x x
momorphismdefineby f (r)= f(rx), ∀r∈(N:x).
x
Remark2.1. ([16,p.625])IfL satisfies(γ),thentheclassofL-injectivemodulesisclosedunder
isomorphism.
Theorem 2.2. ([16, Theorem 1.12, p. 625]) If the subclass L of M satisfies (α),(β) and (γ),
theneverymoduleQhasanL-injectiveenvelopewhichisuniqueuptoQ-isomorphism.
Proposition2.3. IfL satisfiescondition(β),theneverydirectsummandofanL-injectivemodule
isalsoL-injective.
Proof. Aneasycheck.
Corollary2.4. LetL satisfyconditions(β)and(γ)andlet{M} beanyfamilyofmodules. If
i i∈I
(cid:76) M isL-injective,thenM isL-injective,∀i∈I.
i∈I i i
3
Proof. SinceM isisomorphictoadirectsummandof(cid:76) M,thisimmediatebyProposition2.3
i i∈I i
andRemark2.1.
TheconverseofCorollary2.4isnottrueingeneral,considerthefollowingexample:
Example2.5. Let{Ti}i∈I beafamilyofringswithunitandletR=∏i∈ITi betheringproductof
thefamily{T} ,whereadditionandmultiplicationaredefinecomponentwise. LetA=(cid:116) T the
i i∈I i∈I i
directsumofT,∀i∈I. Ifeach T isinjective, ∀i∈I andI isinfinite, then Aisadirectsumof
i Ti i R
injectivemodules,but Aisnotitselfinjective,by[18,p. 140]. Hencewehavethat Aisadirect
R R
sumofL-injectivemodules,but AisnotitselfL-injectivewhereL =M.
R
Proposition2.6. Let{M} beanyfamilyofmodules. Then:
i i∈I
(1) if ∏i∈IMi is L-injective and L satisfies conditions (β) and (γ), then Mi is L-injective,
∀i∈I.
(2)ifMiisL-injective,∀i∈IandL satisfiescondition(λ),then∏i∈IMiisL-injective.
Proof. Thisisobvious.
Since ∏i∈IMi =(cid:76)i∈IMi for any family {Mi}i∈I of modules with finite index set I, thus the
followingcorollaryisimmediatelyfromProposition2.6.
Corollary2.7. LetL satisfycondition(λ)andletM} beanyfamilyof L-injectivemodules.
i i∈I
IfIisafiniteset,then(cid:76) M isL-injective.
i∈I i
Remarks2.8.
(i)LetL ⊆L beanonemptysubclassesofM andletQbeanymodule. IfQisL -injective,
2 1 1
thenQisL -injective.
2
(ii)LetL beanonemptysubclassofM andletL ={(M,N,f,Q)∈L |NisadirectsummandofM}.
1
TheneverymoduleisL -injective.
1
(iii)Theconverseof(i)isnottrueingeneral,forexampleletL =M andL ={(M,N,f,Q)∈
1 2
L |N isadirectsummandofM},thenL ⊆L .By(ii)everyR-moduleisL -injectivebut,ifRis
2 1 2
notsemisimpleartinian,thennoteverymoduleisL -injective(clearlyL -injectivity=injectivity).
1 1
NowwewillintroducetheconceptofP-filterasfollows.
Definition2.9. Letℜ={(M,N)|N≤M,M∈R-Mod}andletρ beanonemptysubclassofℜ. We
saythatρ isaP-filterifρ satisfiesthefollowingconditions:
(i)if(M,N)∈ρ andN≤K≤M,then(M,K)∈ρ;
(ii)forallM∈R-Mod,(M,M)∈ρ;
(iii)if(M,N)∈ρ,then(R,(N:x))∈ρ,∀x∈M.
Example2.10. AllofthefollowingsubclassesofℜareP-filters.
(1) ρT ={(M,N)∈ℜ|N ≤M such that M/N ∈T, M∈R-Mod}, where T is a nonempty
classofmodulesclosedundersubmodulesandhomomorphicimages.
(2)ρ =ℜ={(M,N)|N≤M, M∈R-Mod}.
∞
(3)ρ ={(M,N)∈ℜ|Nisτ-denseinM, M∈R-Mod},whereτisahereditarytorsiontheory.
τ
(4)ρ ={(M,N)∈ℜ | N≤Msuchthatr(M/N)=M/N, M∈R-Mod},whererisaleftexact
r
preradical.
(5)ρ ={(M,N)∈ℜ|N isamaximalsubmoduleinMorN=M, M∈R-Mod}.
max
(6)ρ ={(M,N)∈ℜ|N≤ M, M∈R-Mod}.
e e
It is clear that the P-filters from (2) to (5) are special cases of P-filter in (1). Also, if ρ is a
P-filterthenthesubclassρ ={(R,I)∈ρ |IisaleftidealofR}ofℜisalsoP-filter.
R
Throughoutthischapterwewillfixthefollowingnotations.
-ForanytwoP-filtersρ andρ ,wewilldenotebyL thesubclassL ={(M,N,f,Q)∈
1 2 (ρ1,ρ2) (ρ1,ρ2)
M |M,N,Q∈R-Mod,(M,N)∈ρ and f ∈Hom (N,Q)suchthat(M,ker(f))∈ρ }.
1 R 2
-ForanytwononemptyclassesofmodulesT andF,wewilldenotebyL(T,F) thesubclass
L(T,F)={(M,N,f,Q)∈M |M,N,Q∈R-Mod,N≤MsuchthatM/N∈T and f ∈HomR(N,Q)
withM/ker(f)∈F}. ItisclearthatL(T,F)=L(ρT,ρF),whenT andF areclosedundersub-
modulesandhomomorphicimages.
-Foranytwopreradicalsrands,wewilldenotebyL thesubclassL ={(M,N,f,Q)∈
(r,s) (r,s)
M | M,N,Q∈R-Mod, N≤Msuchthatr(M/N)=M/Nand f ∈ Hom (N,Q)withs(M/ker(f))=
R
M/ker(f)}. ItisclearthatL =L ,whenrandsareleftexactpreradicals.
(r,s) (ρr,ρs)
4
- For any torsion theory τ, we will denote by L the subclass L = {(M,N,f,Q) ∈ M |
τ τ
M,N,Q∈R-Mod,N isaτ-denseinMand f ∈Hom (N,Q)}.ItisclearthatL =L ,when
R τ (ρτ,ρ∞)
τ isahereditarytorsiontheory.
Lemma2.11. Letρ andρ betwoP-filters. ThenL satisfiesconditions(α),(δ)and(µ).
1 2 (ρ1,ρ2)
Proof. Conditions(α)and(δ)areclear.
Condition(µ):Let(M,N,f,Q)∈L andletx∈M,thus(M,N)∈ρ and f ∈Hom (N,Q)
(ρ1,ρ2) 1 R
such that (M,ker(f))∈ρ . Since ρ is a P-filter, thus (R,(N :x))∈ρ . It is easy to prove that
2 1 1
ker(f )=(ker(f):x).Since(M,ker(f))∈ρ andρ isaP-filter,(R,(ker(f):x))∈ρ andhence
x 2 2 2
(R,ker(f ))∈ρ and this implies that (R,(N :x),f ,Q)∈L . Therefore L satisfies
x 2 x (ρ1,ρ2) (ρ1,ρ2)
condition(µ).
One well-known result concerning injective modules states that an R-module M is injective if
andonlyifeveryhomomorphismfromaleftidealofRtoMextendstoahomomorphismfromRto
M ifandonlyifforeachleftidealI ofRandevery f ∈Hom (I,M), thereisanm∈M suchthat
R
f(r)=rm,∀r∈I.ThisisknownasBaer’scondition[4]. Baer’sresultshowsthattheleftidealsof
Rformatestsetforinjectivity.
The following theorem is the main result in this section in which we give a version of Baer’s
criterionforL-injectivity.
Theorem2.12. (GeneralizedBaer’sCriterion)ConsiderthefollowingthreeconditionsforanR-
moduleM:
(1)MisL-injective;
(2)everydiagram
(cid:31)(cid:127) i (cid:47)(cid:47)
I R
f
(cid:15)(cid:15)
M
with(R,I,f,M)∈L canbecompletedtoacommutativediagram;
(3)foreach(R,I,f,M)∈L,thereexistsanelementm∈Msuchthat f(r)=rm, ∀r∈I.
Then (2) and (3) are equivalent and (1) implies (2). Moreover, if L satisfies conditions (α)
and(µ),thenallthethreeconditionsareequivalent.
Proof. (1)⇒(2)and(2)⇔(3)areobvious.
(2)⇒(1)LetL satisfyconditions(α)and(µ)andconsiderthefollowingdiagram
(cid:31)(cid:127) i (cid:47)(cid:47)
A B
f
(cid:15)(cid:15)
M
with(B,A,f,M)∈L.LetS={(C,ϕ)|A≤C≤B,ϕ∈Hom (C,M)suchthat(ϕ(cid:22)A)= f}.Define
R
onSapartialorder(cid:22)by
(C ,ϕ )(cid:22)(C ,ϕ )⇐⇒C ≤C and(ϕ (cid:22)C )=ϕ
1 1 2 2 1 2 2 1 1
Clearly,S(cid:54)=0/ since(A,f)∈S. Furthermore,onecanshowthatSisinductiveinthefollowing
manner. Let F ={(A,f)|i∈I} be an ascending chain in S. Let A =∪ A. Then for any
i i ∞ i∈I i
a∈A thereisa j∈I suchthata∈A ,andsowecandefine f :A →M,by f (a)= f (a). Itis
∞ j ∞ ∞ ∞ j
straightforwardtocheckthat f iswelldefinedand(A ,f )isanupperboundforF inS. Thenby
∞ ∞ ∞
Zorn’sLemma,Shasamaximalelement,say(B(cid:48),g(cid:48)). WewillprovethatB(cid:48)=B.
Supposethatthereexistsx∈B\B(cid:48).Itisclearthat(B,A,f,M)(cid:22)(B,B(cid:48),g(cid:48),M).Since(B,A,f,M)∈
L and L satisfies condition (α), thus (B,B(cid:48),g(cid:48),M)∈L. Since L satisfies condition (µ), thus
(R,(B(cid:48):x),g(cid:48),M)∈L.Byhypothesis, thereexistsahomomorphismg:R→M suchthatg(r)=
x
g(cid:48)(r)=g(cid:48)(rx), ∀r ∈(B(cid:48) :x). Define ψ :B(cid:48)+Rx→M by ψ(b+rx)=g(cid:48)(b)+g(r), ∀b∈B(cid:48),
x
∀r∈R.ψ is a well-defined mapping, since if b +r x=b +r x where b ,b ∈B(cid:48) and r,r ∈R,
1 1 2 2 1 2 2
then(r -r )x=b -b ∈B(cid:48)andhencer -r ∈(B(cid:48):x).Sinceg(cid:48)((r -r )x)=g(r -r )=g(r )-g(r )and
1 2 2 1 1 2 1 2 1 2 1 2
g(cid:48)((r -r )x)=g(cid:48)(b -b )=g(cid:48)(b )-g(cid:48)(b ) thus g(cid:48)(b )+g(r )=g(cid:48)(b )+g(r ) and this implies that
1 2 2 1 2 1 1 1 2 2
ψ(b +r x)=ψ(b +r x).Thusψisawell-definedmapping.Itisclearthatψisahomomorphism
1 1 2 2
5
and(B(cid:48),g(cid:48))(cid:22)(B(cid:48)+Rx,ψ). Since(B(cid:48)+Rx,ψ)∈SandB(cid:48)(cid:36)B(cid:48)+Rx,thuswehaveacontradiction
to maximality of (B(cid:48),g(cid:48)) in S. Hence B(cid:48) =B and this means that there exists a homomorphism
g(cid:48):B→Msuchthat(g(cid:48)(cid:22)A)= f. ThusMisL-injective.
ThefollowingcorollaryisageneralizationofBaer’sresultin[4],[23,Proposition2.1,p. 201],
[16,Baer’sLemma2.2,p. 628]and[5,Theorem2.4,p. 319].
Corollary2.13. Letρ andρ beanytwoP-filters. Thenthefollowingconditionsareequivalent
1 2
forR-moduleM:
(1)MisL -injective;
(ρ1,ρ2)
(2)everydiagram
(cid:31)(cid:127) i (cid:47)(cid:47)
I R
f
(cid:15)(cid:15)
M
with(R,I,f,M)∈L ,canbecompletedtoacommutativediagram;
(ρ1,ρ2)
(3)foreach(R,I,f,M)∈L ,thereexistsanelementm∈Msuchthat f(r)=rm,∀r∈I.
(ρ1,ρ2)
Proof. ByLemma2.11andTheorem2.12.
ThefollowingcharacterizationofL-injectivityisageneralizationof[22,Proposition1.4,p. 3]
and[9,Proposition2.1.3,p. 53].
Proposition2.14. ConsiderthefollowingthreeconditionsforR-moduleM:
(1)QisL-injective;
(2)everydiagram
(cid:31)(cid:127) i (cid:47)(cid:47)
N M
f
(cid:15)(cid:15)
Q
with(M,N,f,Q)∈L andN≤ M,canbecompletedtoacommutativediagram;
e
(3)everydiagram
(cid:31)(cid:127) i (cid:47)(cid:47)
I R
f
(cid:15)(cid:15)
Q
with(R,I,f,Q)∈L andI≤ R,canbecompletedtoacommutativediagram.
e
Then (1) implies (2), (2) implies (3) and, if L satisfies conditions (α) and (µ), then (3)
implies(1).
Proof. LetL satisfy(α)and(µ)andconsiderthefollowingdiagram
(cid:31)(cid:127) i (cid:47)(cid:47)
I R
f
(cid:15)(cid:15)
Q
with (R,I,f,Q)∈L. Let Ic be a complement left ideal of I in R and let C =I⊕Ic. Thus by
[2, Proposition 5.21, p. 75],C≤ R. Define g:C=I⊕Ic →Q by g(a+b)= f(a), ∀a∈I and
e
∀b∈Ic. It is clear that g is a well-defined homomorphism and (R,I,f,Q)(cid:22)(R,C,g,Q). Since
L satisfies condition (α), thus (R,C,g,Q)∈L. By hypothesis, there exists a homomorphism
h:R→Qsuchthat(h(cid:22)C)=g.Thus(h(cid:22)I)=(g(cid:22)I)= f andthisimpliesthatQisL-injective,
byTheorem2.12.
In the following theorem we extend a characterization due to [24, Theorem 2, p. 8] of L-
injectivemodulesovercommutativeNoetherianrings.
6
Theorem2.15. LetRbeacommutativeNoetherianring, letM beanR-moduleandsupposethat
L satisfiesconditions(α)and(µ). ThenMisL-injectiveifandonlyifeverydiagram
(cid:31)(cid:127) i (cid:47)(cid:47)
I R
f
(cid:15)(cid:15)
M
with(R,I,f,M)∈L andIisaprimeidealofR,canbecompletedtoacommutativediagram.
Proof. (=⇒)Thisisobvious.
(⇐=)Considerthefollowingdiagram
(cid:31)(cid:127) i (cid:47)(cid:47)
A B
f
(cid:15)(cid:15)
M
with (B,A,f,M)∈L. Let S={(C,ϕ)|A≤C≤B,ϕ ∈Hom (C,M) such that (ϕ (cid:22)A)= f }.
R
DefineonSapartialorder(cid:22)by
(C ,ϕ )(cid:22)(C ,ϕ )⇐⇒C ≤C and(ϕ (cid:22)C )=ϕ
1 1 2 2 1 2 2 1 1
AsintheproofofTheorem2.12,wecanprovethatShasamaximalelement,say(B(cid:48),g(cid:48)). We
willprovethatB(cid:48)=B. Supposethatthereexistsx∈B\B(cid:48). By[24,Theorem1,p. 8],thereexistsan
elementr ∈Rsuchthat(B(cid:48):r x)isaprimeidealinRandr x∈/B(cid:48). Itisclearthat(B,A,f,M)(cid:22)
0 0 0
(B,B(cid:48),g(cid:48),M). Since(B,A,f,M)∈L andL satisfiescondition(α),thus(B,B(cid:48),g(cid:48),M)∈L.Since
L satisfies condition (µ), thus (R,(B(cid:48) :b),g(cid:48),M)∈L,∀b∈B. Put y=r x, thus y∈B\B(cid:48) and
b 0
hence(R,(B(cid:48):y),g(cid:48),M)∈L. Thuswehavethefollowingdiagram
y
(B(cid:48):y)(cid:31)(cid:127) i (cid:47)(cid:47)R
g(cid:48)y (cid:15)(cid:15) (cid:124)(cid:124) g
M
with(R,(B(cid:48):y),g(cid:48),M)∈L and(B(cid:48):y)isaprimeidealinR. Byhypothesis,thereexistsahomo-
y
morphism g:R→M such that g(r)=g(cid:48)(r)=g(cid:48)(ry), ∀r∈(B(cid:48) :y). Define ψ :B(cid:48)+Ry→M by
y
ψ(b+ry)=g(cid:48)(b)+g(r),∀b∈B(cid:48),∀r∈R.AsintheproofofTheorem2.12,wecanprovethatψ is
awell-definedhomomorphismand(B(cid:48),g(cid:48))(cid:22)(B(cid:48)+Ry,ψ). Since(B(cid:48)+Ry,ψ)∈SandB(cid:48)(cid:36)B(cid:48)+Ry,
thuswehaveacontradictiontomaximalityof(B(cid:48),g(cid:48))inS. HenceB(cid:48)=Bandthismeanthatthere
existsahomomorphismg(cid:48):B→Msuchthat(g(cid:48)(cid:22)A)= f. ThusMisL-injective.
Corollary2.16. Letρ andρ beanytwoP-filtersandletRbeacommutativeNoetherianring,
1 2
letMbeanR-module. ThenMisL -injectiveifandonlyifeverydiagram
(ρ1,ρ2)
(cid:31)(cid:127) i (cid:47)(cid:47)
I R
f
(cid:15)(cid:15)
M
with(R,I,f,M)∈L andIisaprimeidealofR,canbecompletedtoacommutativediagram.
(ρ1,ρ2)
Proof. ByLemma2.11andTheorem2.15.
Corollary 2.17. ([24, Theorem 2, p. 8]) Let R be a commutative Noetherian ring, let M be an
R-module. ThenMisinjectiveifandonlyifeverydiagram
(cid:31)(cid:127) i (cid:47)(cid:47)
I R
f
(cid:15)(cid:15)
M
withIisaprimeidealofR,canbecompletedtoacommutativediagram.
Proof. BytakingthetwoP-filtersρ =ρ =ℜandapplyingCorollary2.16.
1 2
7
3 L-M-Injectivity and s-L-M-Injectivity
InthissectionweintroducetheconceptsofL-M-injectivemodulesands-L-M-injectivemod-
ulesasgeneralizationsofM-injectivemodulesandgivesomeresultsaboutthem.
Definition3.1. LetM,Q∈R-Mod. AmoduleQissaidtobeL-M-injective,ifeverydiagram
(cid:31)(cid:127) i (cid:47)(cid:47)
N M
f
(cid:15)(cid:15)
Q
with (M,N,f,Q)∈L, can be completed to a commutative diagram. A module Q is said to be
L-quasi-injective,ifQisL-Q-injective.
Remarks3.2.
(1) It is clear that an R-module Q is L-injective if and only if Q is L-M-injective for all
M∈R-Mod.
(2)IfL satisfiesconditions(α)and(µ),thenbyTheorem2.12wehavethatanR-moduleQis
L-injectiveifandonlyifQisL-R-injective.
(3)Asaspecialcaseof(2),wehavethatforanytwoP-filtersρ andρ ,thenanR-moduleQ
1 2
isL -injectiveifandonlyifQisL -R-injective.
(ρ1,ρ2) (ρ1,ρ2)
For any nonempty subclass L of M and for any R-module M, we will denote by L the
M
subclassL ={(B,A,f,Q)∈L | B=M}.
M
Remark3.3. LetM,Q∈R-Mod. ThenQisL-M-injectiveifandonlyifQisL -injective.
M
It is clear that if L satisfies condition (α), then L does also. Thus, immediately from Re-
M
mark3.3and[16,Theorem1.3,p. 623],wehavethefollowingcorollary.
Corollary3.4. LetM,Q∈R-ModandletL satisfycondition(α). Thenthefollowingfivecondi-
tionsareequivalent:
(1)QisL-M-injective;
(2)QisadirectsummandineachextensionN⊇QsuchthatN⊆Q+rL (E(N),Q);
M
(3)rL (Q)⊆Q;
M
(4)everydiagram
(cid:31)(cid:127) i (cid:47)(cid:47)
A B
f
(cid:15)(cid:15)
Q
withB⊆A+rL (E(B),A,f,Q),canbecompletedtoacommutativediagram;
M
(5)sL (Q)⊆Q.
M
LetM,Q∈R-Mod,itiswell-knownthatamoduleQisM-injectiveifandonlyif f(M)≤Q,for
everyhomomorphism f :E(M)→E(Q)[20,Lemma1.13,p. 7].
ForananalogousresultforL-M-injectivitywefirstfixthefollowingcondition.
(EL): Let L be a subclass of M. Then a module M satisfies condition (EL) if M has an
L-injectiveenvelopewhichisuniqueuptoM-isomorphismand(EL(M),N,f,Q)∈L whenever
(M,N,f,Q)∈L.
Thenexttheoremisthefirstmainresultinthissectioninwhichwegiveageneralizationof[20,
Lemma1.13,p. 7]and[8,Theorem2.1,p. 34].
Theorem 3.5. Let M,Q∈R-Mod and let L satisfy conditions (α), (β) and (γ). Consider the
followingtwoconditions
(1)QisL-M-injective.
(2) f(M)≤Q, for all f ∈HomR(EL(M),EL(Q)) with (M,L,f(cid:22)L,Q)∈L where L={m∈
M | f(m)∈Q}=M(cid:84)f−1(Q).
Then(1)implies(2)and,ifMsatisfiescondition(EL),then(2)implies(1).
8
Proof. (1)⇒(2) Let f ∈HomR(EL(M),EL(Q) with (M,L,f(cid:22)L,Q)∈L where L={m∈M |
f(m)∈Q}=M(cid:84)f−1(Q). Defineg:L→Qbyg(a)= f(a), ∀a∈L. (i.e., g= f(cid:22)L). Itisclear
thatgisahomomorphism. Thuswehavethefollowingdiagram
(cid:31)(cid:127) i (cid:47)(cid:47)
L M
g
(cid:15)(cid:15) (cid:127)(cid:127) h
Q
with(M,L,g,Q)∈L.ByL-M-injectivityofQ,thereexistsahomomorphismh:M→Qsuchthat
(h(cid:22)L)=g.WewillprovethatQ(cid:84)(f-h)(M)=0.Letx∈Q(cid:84)(f-h)(M),thusthereexistsm∈Msuch
thatx=(f-h)(m)= f(m)-h(m)∈Q. Thus f(m)=x+h(m)∈Qandthisimpliesthatm∈L. Thus
f(m)=g(m)=h(m)andhencex=0. ThusQ(cid:84)(f-h)(M)=0. SinceQisanessentialsubmodule
ofEL(Q)(by[16,Theorem1.19,p. 627])andsince(f-h)(M)≤EL(Q),thus(f-h)(M)=0and
thisimpliesthat f(M)=h(M)≤Q.
(2)⇒(1)LetMsatisfycondition(EL)andconsiderthefollowingdiagram:
(cid:31)(cid:127) i (cid:47)(cid:47)
N M
f
(cid:15)(cid:15)
Q
with (M,N,f,Q)∈L. Since M satisfies condition (EL), thus (EL(M),N,f,Q)∈L. Since L
satisfiescondition(β),thus(EL(M),N,if,EL(Q))∈L. Thuswehavethefollowingdiagram
(cid:31)(cid:127) i (cid:47)(cid:47)
N EL(M)
f
(cid:15)(cid:15)
Q
h
i
(cid:15)(cid:15) (cid:3)(cid:3)
EL(Q)
with(EL(M),N,if,EL(Q))∈L. ByL-injectivityofEL(Q), thereexistsahomomorphismh:
EL(M)→EL(Q)suchthath(n)= f(n), ∀n∈N. LetL={m∈M | h(m)∈Q}. Wewillprove
that (M,L,g,Q)∈L, where g=h(cid:22)L. Let x∈N, thus h(x)= f(x)∈Q and hence x∈L. Thus
N ≤L and (g(cid:22)N)= f. Thus (M,N,f,Q)(cid:22)(M,L,g,Q). Since L satisfies condition (α), thus
(M,L,g,Q)∈L. Byhypothesis,wehavethath(M)≤Qandhenceh(cid:48)=h(cid:22)M:M→Qissuchthat
(h(cid:48)(cid:24)N)= f. ThusQisanL-M-injectivemodule.
Corollary3.6. LetM,Q∈R-Modandletρ andρ beanytwoP-filters. IfM satisfiescondition
1 2
(EL ),thenthefollowingtwoconditionsareequivalent.
(ρ1,ρ2)
(1)QisL -M-injective;
(ρ1,ρ2)
(2) f(M)≤Q, for all f ∈HomR(EL (M),EL (Q)) with (M,L,f(cid:22)L,Q)∈L where
(ρ1,ρ2) (ρ1,ρ2)
L={m∈M | f(m)∈Q}=M(cid:84)f−1(Q).
Proof. ByLemma2.11andTheorem3.5.
Thefollowinglemmaiseasilyproved.
Lemma3.7. IfL =M,thenthefollowingconditionsaresatisfied.
(1)AmoduleMisinjectiveifandonlyifMisL-injective.
(2)EL(M)=E(M),forallM∈R-Mod.
(3)Everymodulesatisfiescondition(EL).
Lemma3.8. Ifτ isahereditarytorsiontheory,thenthefollowingconditionsaresatisfied.
(1)AmoduleMisτ-injectiveifandonlyifMisL -injective.
τ
(2)ELτ(M)=Eτ(M),forallM∈R-Mod.
(3)Everymodulesatisfiescondition(EL ).
τ
(4) For any M,Q∈R-Mod, if f ∈HomR(ELτ(M),ELτ(Q)), then (M,L,f(cid:22)L,Q)∈Lτ, where
L={m∈M | f(m)∈Q}.
9
Proof. (1),(2)and(3)areclear.
(4) Let M,Q∈R-Mod and let f ∈HomR(ELτ(M),ELτ(Q)). Let L={m∈M | f(m)∈Q}.
Define g:(M/L)→(E (Q)/Q) by g(m+L)= f(m)+Q, ∀m∈M. It is clear that g is a well-
τ
defined R-monomorphism. Since Q is a τ-dense in EL (Q)), (EL (Q)/Q) is a τ-torsion. Since
τ τ
g(M/L)≤(EL (Q)/Q)and(M/L)(cid:39)g(M/L),thusM/Lisaτ-torsion. HenceLisaτ-denseinM.
τ
Thus(M,L,f(cid:22)L,Q)∈L .
τ
In the special case L =M is the result [20, Lemma 1.13, p. 7] mentioned earlier and the
followingcorollaryisageneralizationof[8,Theorem2.1,p. 34].
Corollary3.9. LetM,Q∈R-Modandletτ beanyhereditarytorsiontheory. Thenthefollowing
conditionsareequivalent.
(1)Qisτ-M-injective.
(2)QisL -M-injective.
τ
(3) f(M)≤Q,foreveryhomomorphism f :E (M)→E (Q).
τ τ
Proof. ByLemma2.11,Lemma3.8andTheorem3.5.
LetM,Q∈R-Modandletτ beanyhereditarytorsiontheory. AmoduleQiss-τ-M-injectiveif,
foranyN ≤M, anyhomomorphismfromaτ-densesubmoduleofN toQextendstoahomomor-
phismfromN toQ[7,Definition14.6,p. 65].
Asageneralizationofs-τ-M-injectivityandhenceofM-injectivityweintroducetheconceptof
s-L-M-injectivityasfollows.
Definition 3.10. Let M,Q∈R-Mod. A module Q is said to be s-L-M-injective if Q is L-N-
injective,forallN≤M. AmoduleQissaidtobes-L-quasi-injectiveifQiss-L-Q-injective.
Remarks3.11.
(1)Itisclearthateverys-L-M-injectivemoduleisL-M-injective.
(2)LetM,Q∈R-Mod. Thenthefollowingconditionsareequivalent.
(a)Qiss-L-M-injective.
(b)Qiss-L-N-injective,forallN≤M.
(c)QisL-N-injective,forallN≤M.
Fuchs in [14] has obtained a condition similar to Baer’s Criterion that characterizes quasi-
injectivemodules,Blandin[6]hasgeneralizedthattos-τ-quasi-injectivemodules,andCharalam-
bidesin[7]hasgeneralizedthattos-τ-M-injectivemodules.
Ournextaimistogeneralizetheconditiononceagaininordertocharacterizes-L-M-injective
modules. Webeginwiththefollowingcondition.
(L): Let L be a subclass of M and let M be a module . Then M satisfies condition (L)
if for every (B,A,f,Q)∈L, then (Rm,(A:x)m,f ,Q)∈L, for all m∈M and x∈B with
(x,m)
ann (m)⊆(ker(f):x), where f :(A:x)m→Q is a well-defined homomorphism defined by
R (x,m)
f (rm)= f(rx),forallr∈(A:x).
(x,m)
AsubclassL ofM issaidtobefullysubclassifeveryR-modulesatisfiescondition(L).
Example3.12. AllofthefollowingsubclassesofM arefullysubclasses.
(1)L whereT andF arenonemptyclassesofmodulesclosedundersubmodulesandho-
(T,F)
momorphicimages.
(2)L =M.
(3)L ,whereτ isahereditarytorsiontheory.
τ
(4)L ,whereρ andσ areleftexactpreradicals.
(ρ,σ)
Proof. (1) Let (B,A,f,Q)∈L and let m∈M,x∈B such that ann (m)⊆(ker(f):x). By
(T,F) R
Lemma2.11wehavethat L satisfiescondition(µ)andthisimpliesthat(B,(A:x),f ,Q)∈
(T,F) x
L .Itisclearthat(R/(Im:m))(cid:39)(Rm/Im)and(R/(Jm:m))(cid:39)(Rm/Jm)whereI=(A:x)and
(T,F)
J=ker(f ). SinceL =L ,whereρ andρ definedbyρ ={(M,N)∈ℜ|N≤Msuch
x (T,F) (ρ1,ρ2) 1 2 1
thatM/N∈T,M∈R-Mod}andρ ={(M,N)∈ℜ|N≤MsuchthatM/N∈F,M∈R-Mod},thus
2
(R,I)∈ρ and(R,J)∈ρ . SinceI≤(Im:m)≤RandJ≤(Jm:m)≤Randρ ,ρ areP-filters(by
1 2 1 2
example2.10),thus(R,(Im:m))∈ρ and(R,(Jm:m))∈ρ andthisimpliesthat(R/(Im:m))∈T
1 2
and(R/(Jm:m))∈F.SinceT andFareclosedunderhomomorphicimages,thus(Rm/Im)∈T and
10