Publ. Ma . 46 (2002), 453–463
ℵ-PRODUCTS OF MODULES AND SPLITNESS
Feng Lianggui
Abs ac
Le
0−→
ℵ
I
Mα
λ
−→
I
Mα
γ
−→ Coke λ−→ 0
be an exac sequence o modules, in which ℵis an infini e ca dinal,
λ he na u al injec ion and γ he na u al su jec ion. In his pape ,
he condi ions a e gi en mainly in he ou heo ems so ha λ(γ
espec i ely) is spli o locally spli . Consequen ly, some known
esul s a e gene alized. In pa icula , Theo em 1 o [7] and The-
o em 1.6 o [5] a e imp o ed.
1. In oduc ion
Le ℵbe an infini e ca dinal numbe , and {Mi|i∈I}a amily o
le R-modules. As a gene aliza ion o he di ec sum o modules, he
ℵ-p oduc o {Mi|i∈I}is he submodule ℵ
IMi={x∈IMi|
|supp x|<ℵ} ≤ IMi, in which supp xis he suppo se o x=
(xα)α∈I, i.e., supp x={α∈I|xα=0}. So, gi en any amily o le
R-modules {Mα}α∈I, we can always ob ain he ollowing exac sequence:
0−−−−→
ℵ
I
Mα
λ
−−−−→
I
Mα
γ
−−−−→Coke λ−−−−→0,
whe e λdeno es he na u al injec ion and γdeno es he na u al p ojec-
ion. Jus like he di ec sum is no a summand o he di ec p oduc in
gene al, he same case o en happens o he ℵ-p oduc o modules. In
o he wo ds, he na u al injec ion λdoes no spli gene ally. He e, he
ques ions a ise na u ally:
2000 Ma hema ics Subjec Classifica ion. P ima y: 16A50; Seconda y: 16A38.
Key wo ds. ℵ-p oduc , spli ness, ℵ-ACC.
The au ho is suppo ed in pa by Chinese Minis y o Educa ion and NSF o China.
Also, he au ho would like o hank P o . T. Y. Lam and Ma hema ics Depa men
o Uni e si y o Cali o nia, Be keley, o hei help and hospi ali ies.
454 F. Lianggui
Wha condi ions can make λ(γ, esp.) spli , e en a bi weakly, locally
spli ? On he o he hand, assume λis spli , hen wha in o ma ion can
we also ob ain om ha ?
In he p esen pape , we answe he ques ions abo e, and as an ap-
plica ion o he main esul s o his pape , some known esul s a e gen-
e alized. Fo ins ance, we imp o e Theo em 1 o [7] and Theo em 1.6
o [5].
As usual, ings a e associa i e wi h 1 = 0, modules a e uni a y
h oughou his pape . A ca dinal ℵis said o be egula i i is no
o o m o i∈Iµiwi h µi<ℵand |I|<ℵ.
2. Main esul s
Recall ha a le R-module Msa isfies he ℵ-ACC on annihila o s, i
any well-o de ed ascending chain o annihila o s o subse s o Mhas <ℵ
dis inc elemen s. We fi s s a e an equi alen cha ac e iza ion o Mhas
ℵ-ACC on annihila o s.
Lemma 1. Le Mbe a le R-module, ℵan infini e egula ca dinal and
Ian index se wi h |I|=ℵ. Then he ollowing a e equi alen .
(1) Mhas ℵ-ACC on annihila o s;
(2) The na u al map HomR(R/A, IM)−→ HomR(R/A, Coke λ)is
on o o e e y cyclic R-modules R/A, wi h Aa le ideal gene a ed
by ℵelemen s.
P oo : (1) ⇒(2). We only need use (1) ⇒(3) o Theo em 8 o [6].
(2) ⇒(1). Le ωℵdeno e he leas o dinal numbe wi h ca dinali y ℵ,
we can iden i y Iwi h he se o o dinals <ω
ℵ. Suppose Mdoes no
ha e ℵ-ACC on annihila o s, hen by (1) ⇐⇒ (4) o Theo em 8 o [6]
again, we ha e se s
S={mα,α<ω
ℵ}⊆Mand
S={ α,α<ω
ℵ}⊆R
such ha αmα= 0 o all α<ω
ℵand βmα= 0 o all α>β. Take
x=(mα)α<ωℵ, hen x∈IM. Fo all β<ω
ℵ, βx=( βmα)α<ωℵ∈
ℵ
IMbecause βmα= 0 o all β<α. Conside he le ideal Ao
Rgene a ed by { α,α<ω
ℵ}, and le :R−→ IM, −→ x, hen
(A)⊆ℵ
IRx. The e o e he e exis s a unique homomo phism ϕsuch
ℵ-P oduc s o Modules and Spli ness 455
ha he ollowing diag am:
0−−−−→
ℵ
I
Mλ
−−−−→
I
Mγ
−−−−→Coke λ−−−−→0
|A
ϕ
0−−−−→Ai
−−−−→R−−−−→R/A −−−−→0
commu es.
By hypo hesis, he e also exis s a homomo phism ϕ1:R/A −→ IM
such ha ϕ=γϕ1, whe e γ ep esen s he na u al mapping. So, using
Theo em 3.1 o [2], we can find a homomo phism ϕ2:R−→ ℵ
IMsuch
ha |A=ϕ2i. Unde his case, i le ϕ2(1)=(yα)α<ωℵ∈ℵ
IM, hen
βyβ= βmβ= 0 o all β<ω
ℵ, husyα= 0 o each α<ω
ℵ. This
con adic s he ac ha ϕ2(1) ∈ℵ
IM.
Theo em 1. Le ℵbe a egula ca dinal, Ian index se wi h |I|=ℵ,
and Ma le R-module, oge he wi h he sho exac sequence
0−−−−→
ℵ
I
Mλ
−−−−→
I
Mγ
−−−−→Coke λ−−−−→0.
I γis locally spli , hen Mhas ℵ-ACC on annihila o s. Mo eo e , i M
is ai h ul, hen Rhas ℵ-ACC on annihila o s.
P oo : Le Abe a le ideal gene a ed by ℵelemen s. Take any
∈HomR(R/A, Coke λ). No e ha R/A is cyclic, and gene a ed
by ¯
1. So le (¯
1) = x∈Coke λ, hen by hypo hesis, he e exis s a
ψ: Coke λ−→ IMsuch ha γψ(x)=x. Thus, le ψ1=ψ , hen
ψ1∈HomR(R/A, IM) and γψ1=γψ = . Using Lemma 1, i
ollows ha Mhas ℵ-ACC on annihila o s. Fu he mo e, suppose M
is ai h ul and γis locally spli , conside {sm :s∈S, m ∈M}, hen
x{sm;s∈S, m ∈M}=0 ⇐⇒ xS = 0. So he le annihila o o Sis
he annihila o o a subse o M. This comple es he whole p oo .
Obse a ion. Take ℵ=ℵ0in he heo em abo e, ob iously ℵ0
IM=
⊕∞
i=1M.Nowi ⊕∞
i=1Mis a di ec summand o ∞
i=1 M, hen he exac
sequence
0−−−−→
∞
i=1
Mλ
−−−−→
∞
i=1
Mγ
−−−−→Coke λ−−−−→0
spli s. O cou se, γis locally spli . So in his case, Theo em 1 abo e
implies ha Mhas ACC on annihila o s. Fu he mo e, i we le Mbe
456 F. Lianggui
a ai h ul le module, hen Theo em 1 abo e shows also ha Rmus
ha e ACC on annihila o s, which is exac ly he Co olla y 2 o [4]. In
pa icula , when ⊕∞
i=1Ris a di ec summand o ∞
i=1 R, we ge ha R
has ACC on annihila o s, a well-known esul .
I is well known ha a ing Ris le cohe en ⇐⇒ IMiis fla o
any amily o igh fla R-modules and many a emp s ha e been made
o gene alize i , mainly by means o di ec o la ge subdi ec p oduc s
o a ious special modules. Fo example, n-cohe en ings, ℵ-cohe en
ings and so on.
Le ℵbe an infini e ca dinal. Acco ding o [5], a le module Mis said
o be ℵ-fini ely gene a ed i o any subse S⊆Msuch ha |S|<ℵ,
he e exis s a .g. submodule No Msuch ha S⊆N; A ing Ris said o
be le ℵ-cohe en i any .g. le ideal Io Ris ℵ-fini ely p esen ed (in he
sense ha , Ihas he ollowing esolu ion: 0 −→ K−→ F−→ I−→ 0,
in which Fis .g. ee and Kis ℵ-fini ely gene a ed). I is also shown
in [5] ha Ris le ℵ-cohe en i and only i ℵ
IRis igh fla o any
index se I(see, [5, Theo em 1.6]). Now, we poin ou ha his esul
can be imp o ed as ollows.
Theo em 2. Le ℵbe an infini e ca dinal, Ia se wi h |I|=ℵ. Then
he ollowing a e equi alen .
(1) Ris a le ℵ-cohe en ing;
(2) Fo any esolu ion o ℵ
I:0−→ Ki
−→ P−→ ℵ
IR−→ 0in
which Pis p ojec i e and i ep esen s he na u al injec ion, iis
locally spli . In o he wo ds, ℵ
IRis igh fla .
P oo : (1) ⇒(2). By [5, Theo em 1.6], ℵ
IRis igh fla . Di ec ly, by [3,
p. 163, Exe cise 38; p. 154, Co olla y 4.86; p. 129, Theo em 4.23], ℵ
IR
is igh fla ⇐⇒ Fo any esolu ion: 0 −→ Ki
−→ P−→ ℵ
IR−→ 0
wi h p ojec i e P,iis locally spli .
(2) ⇒(1). Le Ibe a fini ely gene a ed le ideal o R,sayI=
R 1+···+R n. Then he e is he exac sequence o le R-modules,
0−−−−→Ki
−−−−→Rnp
−−−−→I=R 1+···+R n−−−−→0
whe e p:Rn−→ Iis defined ia ei−→ i(i=1,...,n). We need show
ha Kis ℵ-fini ely gene a ed. Conside he igh R-module homomo -
phism q:R−→ Rn,x−→ ( 1x,...,
nx), and le ER=Rn/Imq. Then
ℵ-P oduc s o Modules and Spli ness 457
ERis .p., and
0−−−−→Hom(ER,R)−−−−→Hom(Rn,R)−−−−→Hom(Imq,R)
θ2
θ
θ1
0−−−−→K−−−−→Rnp
−−−−→R
commu es, whe e θ: Hom(Rn,R)−→ Rndefined by φ−→ (φ(e1),...,
φ(en)) and θ1: Hom(Imq,R)−→ R ia ϕ−→ ϕ(( 1,...,
n)), is a
monomo phism. The e o e E∗= Hom(ER,R)≃K. Now, we iden-
i y Iwi h he se o o dinals <ω
ℵagain. Fo β<ω
ℵ, le {uα,α<β}
be a subse o E∗, conside he map u:E−→ ℵ
IR,e−→ (xα)α<ωℵ,in
which
xα=uα(e),i α<β;
0,i α≥β.
No e ha Eis .p. and ℵ
IRis igh fla , and hence by [3, p. 133,
Theo em 4.32] again, he e exis s a ee R-module Rmand homomo -
phisms :E−→ Rmand w:Rm−→ ℵ
IRsuch ha w =u. Tha is,
we ha e he ollowing commu a i e diag am:
Rm
Eu
w
ℵ
I
R
Suppose {e1,...,e
m}is he basis o Rm, and le pibe he i h coo dinal
p ojec ion om Rm o R esp. Le 1=p1 ,...,
m=pm , hen o any
e∈E,u(e)=w (e)=w(e1 1(e)+···+em m(e)) = w(e1) 1(e)+···+
w(em) m(e). Mo e explici ly, o α<β,uα(e)=xα=(w(e1))α 1(e)+
···+(w(em))α m(e). Thus uα=(w(e1))α 1+···+(w(em))α m. Fu -
he mo e, {uα,α<β}⊆R 1+···+R m, a .g. submodule o E∗, his
shows E∗is ℵ-fini ely gene a ed, and so Kis ℵ-fini ely gene a ed. The
p oo is comple ed.
Co olla y 1. Le ℵbe a egula ca dinal, Ia se wi h |I|=ℵ. Suppose
ℵ
IRis a di ec summand o IRand IRis igh fla , hen Ris
ℵ-cohe en and has ℵ-ACC on annihila o s.
P oo : F om Theo em 1 and Theo em 2, we deduce his co olla y imme-
dia ely.
458 F. Lianggui
F om now on, le ’s ocus on hose p ope ies o single elemen s o
IMsuch ha
0−−−−→
ℵ
I
Mλ
−−−−→
I
Mγ
−−−−→Coke λ−−−−→0
spli s, whe e ℵis a egula ca dinal and Iis any infini e index se
wi h |I|≥ℵ. Take x∈IM, hen x=(xα)α∈I. Cons uc a am-
ily o ideals o R,Γ(x), as ollows:
Γ(x)={PK(x) = annR{xα}α∈K:K⊆Iand |I K|<ℵ}.
Mo i a ed by he concep o ℵ-ACC on annihila o s o modules, we
say Γ(x) has ℵ-ACC i any well-o de ed ascending chain o Γ(x) has <ℵ
dis inc elemen s. Wi h his in hand, we now s a e he ollowing heo em:
Theo em 3. Le ℵbe a egula ca dinal, {Mα}α∈Ia amily o injec i e
le modules wi h |I|≥ℵ.I Γ(x)has ℵ-ACC o any x∈α∈IMα,
hen
0−−−−→
ℵ
I
Mα
λ
−−−−→
I
Mα
γ
−−−−→Coke λ−−−−→0
spli s. In o he wo ds, ℵ
IMαis injec i e.
Fo he p oo o Theo em 3, we fi s need he ollowing lemma.
Lemma 2. Le {Mα}α∈Ibe a amily o modules, ℵa egula ca dinal
and |I|≥ℵ.Fo x∈α∈IMα, le Ix={ ∈R| x ∈ℵ
IMα}.I
Γ(x)has ℵ-ACC, hen he e exis s y∈ℵ
IMαsuch ha ax =ay o
any a∈Ix.
P oo : We asse ha i Γ(x) has ℵ-ACC, hen Γ(x) has a maximum
elemen . O he wise, ake PK1(x)∈Γ(x), since PK1(x) is no he max-
imum elemen , he e exis s PK2(x)∈Γ(x) such ha PK1(x)PK2(x).
In gene al, o an o dinal β<ω
ℵ, assume we ha e ound PKα(x) o all
α<βsuch ha
PK1(x)PK2(x)···PKα(x)
and
PKα(x)PKα+1 (x) when α+1<β.
ℵ-P oduc s o Modules and Spli ness 459
Case 1: I βis an isola ed o dinal, hen β−1<β. Because PKβ−1(x)is
no a maximum elemen , he e exis s PK(x)∈Γ(x) such ha
PKβ−1(x)PK(x).
Le Kβ=K, hen ∀α<β,PKα(x)PKβ(x).
Case 2: I βis a limi ing o dinal, ake K∗=∩α<βKα, hen |I K∗|=
α<β |I Kα|<ℵsince ℵis egula . So PK∗(x)∈Γ(x), and once mo e,
since PK∗(x) is no a maximum one, he e is PT(x)∈Γ(x) such ha
PK∗(x)PT(x). Le Kβ=T, hen o any α<β, we ha e γsuch ha
α<γ<βbecause βis a limi ing o dinal, as a esul ,
PKβ(x)PK∗(x)PKγ(x)PKα(x).
So, in a wo d, we ha e induc i ely defined a sequence {PKα(x):α<ω
ℵ},
which has ob iously ℵdis inc elemen s. I con adic s he ac ha Γ(x)
has ℵ-ACC.
Now suppose PK∗(x) and PK∗∗ (x) a e wo maximum elemen s o
(Γ(x),⊆). No e ha K∗∩K∗∗ ⊆K∗and K∗∩K∗∗ ⊆K∗∗, hen
PK∗
∩K∗∗ (x)∈Γ(x) and PK∗
(x)⊆PK∗∩K∗∗
(x). Thus, PK∗
(x)=PK∗∩K∗∗
(x).
Simila ly, PK∗∗ (x)=PK∗∩K∗∗ (x). This implies, Γ(x) has only one max-
imum elemen . Recalling he p oo o he o egoing asse ion, we also
find ha each elemen o Γ(x) mus be con ained in a maximum elemen .
The e o e, up o now, we can say he e exis s PK0(x)∈Γ(x) such ha
PK(x)⊆PK0(x) o all PK(x)∈Γ(x).
Le ’s conside y=(yα)α∈I, in which
yα=xα,α∈I K0
0,α∈K0.
Ob iously, y∈ℵ
IMα. Fo any a∈Ix, since ax ∈ℵ
IMα, supp ax =
S⊆Isa isfies |supp ax|<ℵ.SoPI S(x)⊆PK0(x) and a∈PI s(x).
Consequen ly, ax =(axα)α∈I=ay. This comple es he p oo o Lem-
ma 2.
460 F. Lianggui
P oo o Theo em 3: Le be a homomo phism om A o ℵ
IMα,
whe e Ais any le ideal o R. No e IMαis injec i e, he e is a
homomo phism φ:R−→ α∈IMαsuch ha he ollowing diag am
0−−−−→
ℵ
I
Mα−−−−→IMα
φ
0−−−−→A−−−−→R
commu es. Suppose x=(xα)α∈I=φ(1), hen Γ(x) has ℵ-ACC by
assump ion. So, using Lemma 2, we can find y∈ℵ
IMα, such ha
ax =ay o all a∈Ix. Ob iously, A⊆Ix, so i we define a homomo -
phism φ1:R−→ ℵ
IMα ia 1 −→ y, hen we ha e φ1|A= immedi-
a ely. This shows ha ℵ
IMαis injec i e. The p oo o Theo em 3 is
comple ed.
As an applica ion o ou Theo em 1 and Theo em 3, we claim ha
he Theo em III o [1] can be ob ained easily as ou nex co olla y, i.e.,
Co olla y 2. Le ℵbe a egula ca dinal. Fo an injec i e le R-mod-
ule M, he ollowing s a emen s a equi alen .
(1) ℵ
IMis injec i e, o any index se I;
(2) ℵ
IMis injec i e, o some index se Iwi h |I|=ℵ;
(3) Mhas ℵ-ACC on annihila o s.
P oo : (1) ⇒(2). Ob ious.
(2) ⇒(1). Since ℵ
IMis injec i e, he exac sequence
0−−−−→
ℵ
I
Mλ
−−−−→
I
Mγ
−−−−→Coke λ−−−−→0
spli s. So, by Theo em 1, Mhas ℵ-ACC on annihila o s.
(3) ⇒(1). We only need conside he case |I|≥ℵ. In ac , i |I|<ℵ,
hen ℵ
IM=IM, o cou se, is injec i e. Now assume Mhas ℵ-ACC
on annihila o s, na u ally Γ(x) has ℵ-ACC o all x∈IM, so (1) is
ob ained by Theo em 3 immedia ely.
ℵ-P oduc s o Modules and Spli ness 461
P oposi ion 1. Le ℵ1,ℵ2be wo infini e ca dinals wi h ℵ1≤ℵ
2,
{Mα}α∈Ia amily o le R-modules o e an infini e se Iwi h |I|≥ℵ
1.
Then we ha e
(1) The exac sequence: 0−→ ℵ1
α∈IMα
λ
−→ ℵ2
α∈IMα
γ
−→ Coke λ−→
0is always a pu e exac sequence. So, i ℵ2
IMαis p ojec i e,
hen λis locally spli .
(2) I 0−→ ℵ1
α∈IMα
λ
−→ ℵ2
α∈IMα
γ
−→ Coke λ−→ 0spli s, hen
0−→ ℵ1
α∈JMα
λ
−→ ℵ2
α∈JMα
γ
−→ Coke λ−→ 0is also spli ,
o all J⊆I. In pa icula , i ℵ1
IMαis a di ec summand o
IMα, hen ℵ1
JMαis also a di ec summand o JMα o all
J⊆I.
P oo : (1) Gi en any .g. le ideal o R: 1,...,
n, we ha e he exac
sequence: 0 −→ 1,...,
n−→R−→ R/ 1,...,
n−→0.
Fo any homomo phism ϕ:R/ 1,...,
n−→Coke λ, i induces
:R−→ ℵ2
IMαand 1: 1,...,
n−→ℵ1
IMαsuch ha he ol-
lowing diag am
0−−−−→ 1,...,
n−−−−→R−−−−→R/ 1,...,
n−−−−→0
1
ϕ
0−−−−→
ℵ1
I
Mα−−−−→
ℵ2
I
Mα−−−−→Coke λ−−−−→0
commu es. Le (1) = (xα)α∈I, 1( 1)=(x(1)
α)α∈I,..., and 1( n)=
(x(n)
α)α∈I, hen |supp 1( 1)|<ℵ1,...,|supp 1( n)|<ℵ1.So
i(xα)α∈I=( ixα)α∈I=(x(i)
α)α∈I
o each i=1,...,n. Consequen ly,
ixα=0,α∈I supp 1( i);
ixα=0,α∈supp 1( i).
Cons uc y=(yα)α∈I, ia
yα=xα,α∈∪
n
i=1 supp 1( i);
0,α/∈∪
n
i=1 supp 1( i).
Since |∪
n
i=1 supp 1( i)|=n
i=1 |supp 1( i)|<ℵ1,y∈ℵ1
IMα.
Define φ:R−→ ℵ1
IMα ia φ(1) = y, hen 1=φ| 1,..., n.By
[2, Theo em 3.1], he na u al map Hom(R/ 1,...,
n,ℵ2
IMα)−→
