Publ. Ma . 46 (2002), 435–440
WHEN IS EACH PROPER OVERRING OF RAN
S(EIDENBERG)-DOMAIN?
Noˆ
omen Ja boui
Abs ac
A domain Ris called a maximal “non-S” sub ing o a field Li
R⊂L,Ris no an S-domain and each domain Tsuch ha R⊂
T⊆Lis an S-domain. We show ha maximal “non-S”sub ings R
o a field La e he in eg ally closed pseudo- alua ion domains
sa is ying dim(R)=1,dim
(R) = 2 and L=q (R).
1. In oduc ion
Th oughou his pape , R→Sdeno es an ex ension o commu a i e
in eg al domains, q (R) he quo ien field o an in eg al domain Rand
.deg[S:R] he anscendence deg ee o q (S) o e q (R). I .deg[S:
R] = 0, we say ha Sis algeb aic o e R. We ecall ha a ing Ro fini e
K ull dimension nis a Jaffa d ing i i s alua i e dimension ( he limi
o he sequence (dim(R[X1,...,X
n]) −n,n∈N)) dim (R), is also n.
P ¨u e domains and Noe he ian domains a e Jaffa d domains. Recall
ha a domain Ris an S-domain [12] i o each heigh 1 p ime ideal p
o R, he ex ended p ime p[X] in one inde e mina e is also heigh 1 in
R[X]. We assume amilia i y wi h hese concep s as in [1] and [12].
In [3], he au ho and M. Ben Nas conside ed maximal non-Jaffa d
sub ings o a field L, ha is, he domains Rwhe e Ris a non Jaffa d
domain and each ing T,R⊂T⊆Lis Jaffa d. They cha ac e ized
hese domains in e ms o pseudo- alua ion domains. On he o he hand
he au ho and I. Yengui in [11] s udied he domains Rsuch ha each
domain con ained be ween Rand i s quo ien field is an S-domain. They
a e said o be absolu ely S-domains. To comple e his ci cle o ideas and
o hono Seidenbe g we deal wi h maximal “non-S” sub ing(s) o a field;
ha is, he domains R, whe e Ris no an S-domain and each ing T,
R⊂T⊆Lis an S-domain. Fi s we show ha i Ris a maximal
2000 Ma hema ics Subjec Classifica ion. P ima y: 13B02; Seconda y: 13C15,
13A17, 13A18, 13B25, 13E05.
Key wo ds. Jaffa d domain, S-domain, alua ion domain, K ull dimension, pullback.
436 N. Ja boui
“non-S” sub ing o a field L, hen L= q (R). Hence, we may es ic
ou sel es o he case whe e L= q (R). Le us ecall some e minology:
Le Tbe a ing, Ian ideal o T,Dbe a sub ing o T/I and le Rbe he
sub ing o Tdefined by he ollowing pullback cons uc ion:
R−−−−→D
T−−−−→T/I
Following [4], we say ha Ris he ing o he (T,I,D)cons uc ion
and we se R:= (T,I,D). No e ha R:= (T,I,D) i and only i is
con ained in Tand sha es he ideal Iwi h he ing T. The (T,I,D)
cons uc ions we e conside ed o he fi s ime in [7], in he con es o
gene al pullback cons uc ion. Pa icula ly he las cons uc ion o be
no ed he e conce ns he no ion o a pseudo- alua ion domain ( o sho ,
a PVD), which was in oduced by J. R. Heds om and E. G. Hous on [9]
and has been s udied subsequen ly in [2], [5], [6] and [10]. A domain R
is said o be a PVD in case each p ime ideal po Ris s ongly p ime,
in he sense ha whene e x, y ∈q (R) sa is y xy ∈p, hen ei he
x∈po y∈p, equi alen ly, in case Rhas a (uniquely de e mined)
alua ion o e ing Vsuch ha Spec(R) = Spec(V) as se s, equi alen ly
(by [2, P oposi ion 2.6]) in case Ris a pullback o he o m V×Kk,
whe e Vis a alua ion domain wi h esidue field Kand kis a subfield
o K. As he e minology sugges s, any alua ion domain is a PVD [9,
P oposi ion 1.1]. Al hough he con e se is alse [9, Example 2.1], any
PVD mus , a leas , be local [9, Co olla y 1.3]. The main esul o his
pape is Theo em 2.2, which s a es ha Ris a maximal “non-S” sub ing
o q (R) i and only i Ris an in eg ally closed pseudo- alua ion domain
wi h dim(R) = 1 and dim (R) = 2. As an applica ion o Theo em 2.2,
we gi e necessa y and sufficien condi ions o ce ain pullbacks o be
maximal “non-S” sub ings o hei quo ien fields.
2. Main esul s
Le Rbe a domain con ained in a field L. We say ha Ris a maximal
“non-S”sub ing o Li Ris no an S-domain and each ing Tsuch ha
R⊂T⊆Lis an S-domain.
Fi s o all, we es ablish he ollowing:
P oposi ion 2.1. Le Rbe a domain and La field con aining R.I R
is a maximal “non-S”sub ing o L, hen L= q (R).
P ope O e ings 437
P oo : Fi s no ice ha Lis algeb aic o e R. Indeed, i no hen he e
exis s an elemen o L anscenden al o e R. Hence each o e ing
o R[ ] should be an S-domain ha is R[ ] is an absolu ely S-domain.
Hence by [11, P oposi ion 1.14] Ris a field which con adic s he ac
ha Ris no an S-domain. Now ou ask is o show ha L= q (R). As-
sume ha q (R)⊂L, and le α∈L q (R). Then αis algeb aic o e R.
Thus he e exis s an elemen ∈Rsuch ha α is in eg al o e R.Thus
R⊂R[ α] is an in eg al ex ension. Bu R[ α] is an S-domain. Hence R
is an S-domain, he desi ed con adic ion o comple e he p oo .
As a di ec consequence o P oposi ion 2.1, he s udy o maximal
“non-S” sub ing(s) o a field Lcan be educed o he case whe e L=
q (R). Now no ice ha i Ris a maximal “non-S”sub ing o q (R), hen
Ris in eg ally closed. Indeed, i R=R, hen Ris an S-domain, and
hence so is R(since R⊂Ris an in eg al ex ension), which is impossible.
Ou main esul is he ollowing:
Theo em 2.2. Le Rbe a domain. Then he ollowing s a emen s a e
equi alen :
(i) Ris a maximal “non-S” sub ing o q (R);
(ii) Ris an in eg ally closed PVD wi h dim(R)=1and dim (R)=2.
P oo : (i) ⇒(ii). We ha e al eady no iced ha Ris in eg ally closed.
On he o he hand since Ris no an S-domain, hen he e is a heigh 1
p ime ideal po Rsuch ha h (p[X]) = 2. Then he e is a nonze o
p ime ideal Po R[X] con ained in p[X] such ha P∩R= (0). Thus
Ris a sub ing o R1=R[X]/P which is isomo phic o R[u], whe e uis
an algeb aic elemen o e R.By[8, Co olla y 19.7], he e is a alua ion
o e ing Wo R1con aining a p ime ideal Po heigh 1 such ha
P∩R1=p[X]/P. Deno ing V=W∩q (R), Vis a alua ion o e ing
o Rcon aining a heigh 1 p ime ideal q=P∩q (R)[8, Theo em 19.16]
such ha q∩R=p.Now, .deg[W/P:V/q]=0[8, Theo em 19.16].
Hence
.deg[V/q :R/p]= .deg[W/P:R/p]
≥ .deg[R1/(p[X]/P ):R/p]
= .deg[(R[X]/P )/(p[X]/P ):R/p]
= .deg[(R[X]/p[X]) : R/p]=1.
438 N. Ja boui
Assume ha R=(Vq,qV
q,R
p/pRp), hen he domain (Vq,qV
q,R
p/pRp)
is a p ope o e ing o Rand i should be an S-domain and by [11,
P oposi ion 1.4], we ge .deg[Vq/qVq:Rp/pRp] = 0 which is impos-
sible. The e o e R:= (Vq,qV
q,R
p/pRp). Hence Ris a PVD (c . [2]).
Ou ask now is o show ha .deg[Vq/qVq:Rp/pRp] = 1. The ex en-
sion Rp/pRp⊂Vq/qVqcan no be algeb aic since Ris no an S-domain
[11, P oposi ion 1.4]. Assume ha .deg[Vq/qVq:Rp/pRp]≥2, and
le X,Ybe wo anscenden al algeb aically independen elemen s o
Vq/qVqo e Rp/pRp. Then he domain T:= (Vq,qV
q,(Rp/pRp)[X]) is
a p ope o e ing o R, husTis an S-domain. Hence by [11, P oposi-
ion 1.4], we ge .deg[Vq/qVq:(Rp/pRp)[X]] = 0, which is impossible.
Hence .deg[Vq/qVq:Rp/pRp] = 1. The e o e by [1, P oposi ion 2.5],
dim(R) = 1 and dim (R)=2.
(ii) ⇒(i). Since Ris a PVD, hen R:= (V,M,k), whe e Vis a
alua ion domain wi h maximal ideal Mand kis a field. I is clea ha
Ris no an S-domain because .deg[V/M :R/M] = 1. Now, le Tbe a
domain such ha R⊂T⊆q (R). Then by [3, Lemma 1.3], ei he Tis
an o e ing o V, so i is an S-domain, o Tis an in e media e domain
be ween Rand V,soT:= (V,M,D), whe e R/M ⊂D⊆V/M. Since R
is in eg ally closed, hen .deg[V/M :D] = 0. Thus Tis an S-domain.
Hence Ris a maximal “non-S” sub ing o q (R).
Now we de e mine when a pullback Ris a maximal “non-S” sub ing
o i s quo ien field. We ecall some no a ion o conduc o s. I Ris
a domain and I,Ja e R-submodules o q (R), hen (I:J)={x∈
q (R)|xJ ⊂I}.I Ris a PVD wi h associa ed alua ion domain Vand
maximal ideal M, assume ha R=V, hen Mis no a p incipal ideal
o Rand V=(M:M)[2, P oposi ion 2.3], and by [2, Lemma 2.4], we
ge V=(R:M)=(M:M).
We es ablish he ollowing heo em.
Theo em 2.3. Le Tbe a domain, Ma maximal ideal o Tand Da
sub ing o he field K=T/M.Le R:= (T,M,D). Then he ollowing
s a emen s a e equi alen :
(i) Ris a maximal “non-S” sub ing o q (R);
(ii) Dis a field algeb aically closed in (M:M)/M , wi h .deg[K:
D]=1and Tis a one-dimensional Jaffa d PVD.
P oo : (i) ⇒(ii). By Theo em 2.2, Ris a PVD. Hence he e exis s a
alua ion domain Vwi h mas a maximal ideal such ha R:= (V, m, k),
whe e kis a field. Since Tis an o e ing o R, hen by [3, Lemma 1.3],
ei he R⊂T⊆Vo V⊆T.
P ope O e ings 439
Case 1: I R⊂T⊆V, hen Tsha es he ideal mwi h Rand V,so
T:= (V, m, T/m). Bu we ha e M⊆m(since Ris local wi h maximal
ideal m). Thus M=mbecause Mis a maximal ideal o T. Hence
T:= (V,M,K), D=R/M =R/m =k,soDis a field. On he o he
hand Ris in eg ally closed (Theo em 2.2), hus Dis algeb aically closed
in V/M =(M:M)/M . We ha e dim(T) = dim(V) = dim(R) = 1, and
since Tis an S-domain, hen dim(T) = dim (T) = 1. Now .deg[K:
D] = dim (R)−dim (T)=1.
Case 2: I Tis an o e ing o V, hen T=Vsince Vis a one-dimensional
alua ion domain. Thus m=M. This yields D=R/M =R/m =k
and i is ob ious ha Dis algeb aically closed in V/M =(M:M)/M .
On he o he hand .deg[K:D] = dim (R)−dim (T)=1.
(ii) ⇒(i). Since D⊂Kis no an algeb aic ex ension, hen Ris no an
S-domain [11, P oposi ion 1.4]. The ing Tis a PVD, so he e is a al-
ua ion domain Wwi h maximal ideal Msuch ha T:= (W, M, K).
Bu R:= (T,M,D). Hence Ris a PVD wi h associa ed alua ion
domain W=(M:M). Fu he mo e, dim(R) = dim(T) = 1 and
dim (R) = dim (T) + dim (D)+ .deg[K:D] = 2. Since Dis al-
geb aically closed in W/M, hen Ris in eg ally closed. Thus by Theo-
em 2.2, Ris a maximal “non-S” sub ing o q (R).
Acknowledgemen . The au ho exp ess hanks o he e e ees o alu-
able sugges ions.
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Depa men o Ma hema ics
Facul y o Sciences o S ax
3018 S ax, BP 802
Tunisia
E-mail add ess:[email p o ec ed]
P ime a e si´o ebuda el 8 de no emb e de 2001,
da e a e si´o ebuda el 16 de maig de 2002.
