arXiv:1610.10023v1 [] 31 Oct 2016

QUASI-PRÜFER EXTENSIONS OF RINGS
GABRIEL PICAVET AND MARTINE PICAVET-L’HERMITTE
Abstract. We introduce quasi-Prüfer ring extensions, in order
to relativize quasi-Prüfer domains and to take also into account
some contexts in recent papers, where such extensions appear in a
hidden form. An extension is quasi-Prüfer if and only if it is an INC
pair. The class of these extensions has nice stability properties.
We also define almost-Prüfer extensions that are quasi-Prüfer, the
converse being not true. Quasi-Prüfer extensions are closely linked
to finiteness properties of fibers. Applications are given for FMC
extensions, because they are quasi-Prüfer.

1. Introduction and Notation
We consider the category of commutative and unital rings. An epimorphism is an epimorphism of this category. Let R ⊆ S be a (ring)
extension. The set of all R-subalgebras of S is denoted by [R, S]. The
extension R ⊆ S is said to have FIP (for the “finitely many intermediate algebras property”) if [R, S] is finite. A chain of R-subalgebras of S
is a set of elements of [R, S] that are pairwise comparable with respect
to inclusion. We say that the extension R ⊆ S has FCP (for the “finite
chain property”) if each chain in [R, S] is finite. Dobbs and the authors
characterized FCP and FIP extensions [10]. Clearly, an extension that
satisfies FIP must also satisfy FCP. An extension R ⊆ S is called FMC
if there is a finite maximal chain of extensions from R to S.
We begin by explaining our motivations and aims. The reader who is
not familiar with the notions used will find some Scholia in the sequel,
as well as necessary definitions that exist in the literature. Knebusch
and Zang introduced Prüfer extensions in their book [26]. Actually,
these extensions are nothing but normal pairs, that are intensively
studied in the literature. We do not intend to give an extensive list of
recent papers, written by Ayache, Ben Nasr, Dobbs, Jaballah, Jarboui
and some others. We are indebted to these authors because their papers are a rich source of suggestions. We observed that some of them
2010 Mathematics Subject Classification. Primary:13B22; Secondary:
Key words and phrases. flat epimorphism, FIP, FCP extension, minimal extension, integral extension, Morita, Prüfer hull, support of a module.
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G. PICAVET AND M. PICAVET

are dealing with FCP (FIP, FMC) extensions, followed by a Prüfer
extension, perhaps under a hidden form. These extensions reminded
us quasi-Prüfer domains (see [18]). Therefore, we introduced in [38]
quasi-Prüfer extensions R ⊆ S as extensions that can be factored
R ⊆ R′ ⊆ S, where the first extension is integral and the second is
Prüfer. Note that FMC extensions are quasi-Prüfer.
We give a systematic study of quasi-Prüfer extensions in Section 2
and Section 3. The class of quasi-Prüfer extensions has a nice behavior
with respect to the classical operations of commutative algebra. An important result is that quasi-Prüfer extensions coincide with INC-pairs.
Another one is that this class is stable under forming subextensions
and composition. A striking result is the stability of the class of quasiPrüfer extensions by absolutely flat base change, like localizations and
Henselizations. Any ring extension R ⊆ S admits a quasi-Prüfer closure, contained in S. Examples are provided by Laskerian pairs, open
pairs and the pseudo-Prüfer pairs of Dobbs-Shapiro [15].
Section 4 deals with almost-Prüfer extensions, a special kind of quasiPrüfer extensions. They are of the form R ⊆ T ⊆ S, where the first
extension is Prüfer and the second is integral. Any ring extension
admits an almost-Prüfer closure, contained in S. The class of almostPrüfer extensions seems to have less properties than the class of quasiPrüfer extensions but has the advantage of the commutation of Prüfer
closures with localizations at prime ideals. We examine the transfer of
the quasi (almost)-Prüfer properties to subextensions.
Section 5 study the transfer of the quasi (almost)-Prüfer properties
to Nagata extensions.
In section 6, we complete and generalize the results of Ayache-Dobbs
in [5], with respect to the finiteness of fibers. These authors have evidently considered particular cases of quasi-Prüfer extensions. A main
result is that if R ⊆ S is quasi-Prüfer with finite fibers, then so is
R ⊆ T for T ∈ [R, S]. In particular, we recover a result of [5] about
FMC extensions.
Now Section 7 gives calculations of |[R, S]| with respect to its Prüfer
closure, quasi-Prüfer (almost-Prüfer) closure in case R ⊆ S has FCP.
1.1. Recalls about some results and definitions. The reader is
warned that we will mostly use the definition of Prüfer extensions by
flat epimorphic subextensions investigated in [26]. The results needed
may be found in Scholium A for flat epimorphic extensions and some
results of [26] are summarized in Scholium B. Their powers give quick
proofs of results that are generalizations of results of the literature.

QUASI-PRÜFER EXTENSIONS

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As long as FCP or FMC extensions are concerned, we use minimal
(ring) extensions, a concept introduced by Ferrand-Olivier [17]. An
extension R ⊂ S is called minimal if [R, S] = {R, S}. It is known that
a minimal extension is either module-finite or a flat epimorphism [17]
and these conditions are mutually exclusive. There are three types of
integral minimal (module-finite) extensions: ramified, decomposed or
inert [36, Theorem 3.3]. A minimal extension R ⊂ S admits a crucial
ideal C(R, S) =: M which is maximal in R and such that RP = SP
for each P 6= M, P ∈ Spec(R). Moreover, C(R, S) = (R : S) when
R ⊂ S is an integral minimal extension. The key connection between
the above ideas is that if R ⊆ S has FCP or FMC, then any maximal
(necessarily finite) chain of R-subalgebras of S, R = R0 ⊂ R1 ⊂ · · · ⊂
Rn−1 ⊂ Rn = S, with length n < ∞, results from juxtaposing n
minimal extensions Ri ⊂ Ri+1 , 0 ≤ i ≤ n − 1.
Following [24], we define the length ℓ[R, S] of [R, S] as the supremum
of the lengths of chains in [R, S]. In particular, if ℓ[R, S] = r, for some
integer r, there exists a maximal chain in [R, S] with length r.
As usual, Spec(R), Max(R), Min(R), U(R), Tot(R) are respectively
the set of prime ideals, maximal ideals, minimal prime ideals, units,
total ring of fractions of a ring R and κ(P ) = RP /P RP is the residual
field of R at P ∈ Spec(R).
If R ⊆ S is an extension, then (R : S) is its conductor and if P ∈
Spec(R), then SP is the localization SR\P . We denote the integral
S
closure of R in S by R (or R).
A local ring is here what is called elsewhere a quasi-local ring. The
support of an R-module E is SuppR (E) := {P ∈ Spec(R) | EP 6= 0}
and MSuppR (E) := SuppR (E) ∩ Max(R). Finally, ⊂ denotes proper
inclusion and |X| the cardinality of a set X.
Scholium A We give some recalls about flat epimorphisms (see [27,
Chapitre IV], except (2) which is [31, Proposition 2]).
(1) R → S is a flat epimorphism ⇔ for all P ∈ Spec(R), either
RP → SP is an isomorphism or S = P S ⇔ RP ⊆ SP is a flat
epimorphism for all P ∈ Spec(R).
(2) (S) A flat epimorphism, with a zero-dimensional domain, is surjective.
(3) If f : A → B and g : B → C are ring morphisms such that g ◦ f
is injective and f is a flat epimorphism, then g is injective.
(4) Let R ⊆ T ⊆ S be a tower of extensions, such that R ⊆ S is a
flat epimorphism. Then T ⊆ S is a flat epimorphism but R ⊆ T
does not need. A Prüfer extension remedies to this defect.

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G. PICAVET AND M. PICAVET

(5) (L) A faithfully flat epimorphism is an isomorphism. Hence,
R = S if R ⊆ S is an integral flat epimorphism.
(6) If f : R → S is a flat epimorphism and J an ideal of S, then
J = f −1 (J)S.
(7) If f : R → S is an epimorphism, then f is spectrally injective
and its residual extensions are isomorphisms.
(8) Flat epimorphisms remain flat epimorphisms under base change
(in particular, after a localization with respect to a multiplicatively closed subset).
(9) Flat epimorphisms are descended by faithfully flat morphisms.
1.2. Recalls and results on Prüfer extensions. We recall some
definitions and properties of ring extensions R ⊆ S and rings R. There
are a lot of characterizations of Prüfer extensions. We keep only those
that are useful in this paper. We give the two definitions that are dual
and emphasize some characterizations in the local case.
Scholium B
(1) [26] R ⊆ S is called Prüfer if R ⊆ T is a flat epimorphism for
each T ∈ [R, S].
(2) R ⊆ S is called a normal pair if T ⊆ S is integrally closed for
each T ∈ [R, S].
(3) R ⊆ S is Prüfer if and only if it is a normal pair [26, Theorem
5.2(4)].
(4) R is called Prüfer if its finitely generated regular ideals are
invertible, or equivalently, R ⊆ Tot(R) is Prüfer [20, Theorem
13((5)(9))].
Hence Prüfer extensions are a relativization of Prüfer rings. Clearly,
a minimal extension is a flat epimorphism if and only if it is Prüfer.
We will then use for such extensions the terminology: Prüfer minimal
extensions. The reader may find some properties of Prüfer minimal
extensions in [36, Proposition 3.2, Lemma 3.4 and Proposition 3.5], asserted by L. Dechene in her dissertation, but where in addition R must
be supposed local. The reason why is that this word has surprisingly
disappeared during the printing process of [36].
We will need the two next results. Some of them do not explicitely
appear in [26] but deserve to be emphasized. We refer to [26, Definition
1, p.22] for a definition of Manis extensions.
Proposition 1.1. Let R ⊆ S be a ring extension.
(1) R ⊆ S is Prüfer if and only if RP ⊆ SP is Prüfer for each
P ∈ Spec(R) (respectively, P ∈ Supp(S/R)).

QUASI-PRÜFER EXTENSIONS

5

(2) R ⊆ S is Prüfer if and only if RM ⊆ SM is Manis for each
M ∈ Max(R).
Proof. (1) The class of Prüfer extensions is stable under localization [26,
Proposition 5.1(ii), p.46-47]. To get the converse, use Scholium A(1).
(2) follows from [26, Proposition 2.10, p.28, Definition 1, p.46].

Proposition 1.2. Let R ⊆ S be a ring extension, where R is local.
(1) R ⊆ S is Manis if and only if S \ R ⊆ U(S) and x ∈ S \ R ⇒
x−1 ∈ R. In that case, R ⊆ S is integrally closed.
(2) R ⊆ S is Manis if and only if R ⊆ S is Prüfer.
(3) R ⊆ S is Prüfer if and only if there exists P ∈ Spec(R) such
that S = RP , P = SP and R/P is a valuation domain. Under
these conditions, S/P is the quotient field of R/P .
Proof. (1) is [26, Theorem 2.5, p.24]. (2) is [26, Scholium 10.4, p.147].
Then (3) is [10, Theorem 6.8].

Next result shows that Prüfer FCP extensions can be described in a
special manner.
Proposition 1.3. Let R ⊂ S be a ring extension.
(1) If R ⊂ S has FCP, then R ⊂ S is integrally closed ⇔ R ⊂ S is
Prüfer ⇔ R ⊂ S is a composite of Prüfer minimal extensions.
(2) If R ⊂ S is integrally closed, then R ⊂ S has FCP ⇔ R ⊂ S is
Prüfer and Supp(S/R) is finite.
Proof. (1) Assume that R ⊂ S has FCP. If R ⊂ S is integrally closed,
then, R ⊂ S is composed of Prüfer minimal extensions by [10, Lemma
3.10]. Conversely, if R ⊂ S is composed of Prüfer minimal extensions,
R ⊂ S is integrally closed, since so is each Prüfer minimal extension. A
Prüfer extension is obviously integrally closed, and an FCP integrally
closed extension is Prüfer by [10, Theorem 6.3].
(2) The logical equivalence is [10, Theorem 6.3].

Definition 1.4. [26] A ring extension R ⊆ S has:
bS , called the
(1) a greatest flat epimorphic subextension R ⊆ R
Morita hull of R in S.
eS , called the Prüfer hull
(2) a greatest Prüfer subextension R ⊆ R
of R in S.
b := R
bS and R
e := R
eS , if no confusion can occur. R ⊆ S is
We set R
e
called Prüfer-closed if R = R.
eS is denoted by P(R, S) in [26] and R
bS is the weakly
Note that R
surjective hull M(R, S) of [26]. Our terminology is justified because

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G. PICAVET AND M. PICAVET

Morita’s work is earlier [30, Corollary 3.4]. The Morita hull can be
computed by using a (transfinite) induction [30]. Let S ′ be the set of
all s ∈ S such that there is some ideal I of R, such that IS = S and
Is ⊆ R. Then R ⊆ S ′ is a subextension of R ⊆ S. We set S1 := S ′
and Si+1 := (Si )′ ⊆ Si . By [30, p.36], if R ⊂ S is an FCP extension,
b = Sn for some integer n.
then R
At this stage it is interesting to point out a result; showing again
that integral closedness and Prüfer extensions are closely related.
Proposition 1.5. Olivier [33, Corollary, p.56] An extension R ⊆ S is
integrally closed if and only if there is a pullback square:
R −−−→


y

S


y

V −−−→ K
where V is a semi-hereditary ring and K its total quotient ring.
In that case V ⊆ K is a Prüfer extension, since V is a Prüfer ring,
whose localizations at prime ideals are valuation domains and K is an
absolutely flat ring. As there exist integrally closed extensions that
are not Prüfer, we see in passing that the pullback construction may
not descend Prüfer extensions. The above result has a companion for
minimal extensions that are Prüfer [21, Proposition 3.2].
Proposition 1.6. Let R ⊆ S be an extension and T ∈ [R, S], then
eT = R
e ∩ T . Therefore, for T, U ∈ [R, S] with T ⊆ U, then R
eT ⊆ R
eU .
R

eT is the greatest Prüfer extenProof. Obvious, since the Prüfer hull R
sion R ⊆ V contained in T .

e if R ⊆ S has FCP.
We will show later that in some cases Te ⊆ U
2. Quasi-Prüfer extensions
We introduced the following definition in [38, p.10].
Definition 2.1. An extension of rings R ⊆ S is called quasi-Prüfer if
one of the following equivalent statements holds:
(1) R ⊆ S is a Prüfer extension;
(2) R ⊆ S can be factored R ⊆ T ⊆ S, where R ⊆ T is integral
and T ⊆ S is Prüfer. In that case R = T

To see that (2) ⇒ (1) observe that if (2) holds, then T ⊆ R is integral
and a flat injective epimorphism, so that R = T by (L) (Scholium A(5)).

QUASI-PRÜFER EXTENSIONS

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We observe that quasi-Prüfer extensions are akin to quasi-finite extensions if we refer to Zariski Main Theorem. This will be explored in
Section 6, see for example Theorem 6.2.
Hence integral or Prüfer extensions are quasi-Prüfer. An extension
is clearly Prüfer if and only if it is quasi-Prüfer and integrally closed.
Quasi-Prüfer extensions allow us to avoid FCP hypotheses.
We give some other definitions involved in ring extensions R ⊆ S.
The fiber at P ∈ Spec(R) of R ⊆ S is FibR,S (P ) := {Q ∈ Spec(S) |
Q ∩ R = P }. The subspace FibR,S (P ) of Spec(S) is homeomorphic
to the spectrum of the fiber ring FR,S (P ) := κ(P ) ⊗R S at P . The
homeomorphism is given by the spectral map of S → κ(P ) ⊗R S and
κ(P ) → κ(P ) ⊗R S is the fiber morphism at P .
Definition 2.2. A ring extension R ⊆ S is called:
(1) incomparable if for each pair Q ⊆ Q′ of prime ideals of S, then
Q ∩ R = Q′ ∩ R ⇒ Q = Q′ , or equivalently, κ(P ) ⊗R T is a
zero-dimensional ring for each T ∈ [R, S] and P ∈ Spec(R),
such that κ(P ) ⊗R T 6= 0.
(2) an INC-pair if R ⊆ T is incomparable for each T ∈ [R, S].
(3) residually algebraic if R/(Q ∩ R) ⊆ S/Q is algebraic for each
Q ∈ Spec(S).
(4) a residually algebraic pair if the extension R ⊆ T is residually
algebraic for each T ∈ [R, S].
The following characterization was announced in [38]. We were unaware that this result is also proved in [7, Corollary 1], when we present
it in ArXiv. However, our proof is largely shorter because we use the
powerful results of [26].
Theorem 2.3. An extension R ⊆ S is quasi-Prüfer if and only if
R ⊆ S is an INC-pair and, if and only if, R ⊆ S is a residually
algebraic pair.
Proof. Suppose that R ⊆ S is quasi-Prüfer and let T ∈ [R, S]. We
set U := RT . Then R ⊆ U is a flat epimorphism by definition of a
Prüfer extension and hence is incomparable as is R ⊆ R . It follows
that R ⊆ U is incomparable. Since T ⊆ U is integral, it has going-up.
It follows that R ⊆ T is incomparable. Conversely, if R ⊆ S is an
INC-pair, then so is R ⊆ S. Since R ⊆ S is integrally closed, R ⊆ S
is Prüfer [26, Theorem 5.2,(9’), p.48]. The second equivalence is [14,
Proposition 2.1] or [18, Theorem 6.5.6].

Corollary 2.4. An extension R ⊆ S is quasi-Prüfer if and only if
R ⊆ T is Prüfer for each T ∈ [R, S].

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G. PICAVET AND M. PICAVET

It follows that most of the properties described in [6] for integrally
closed INC-pairs of domains are valid for arbitrary ring extensions.
Moreover, a result of Dobbs is easily gotten: an INC-pair R ⊆ S is
an integral extension if and only if R ⊆ S is spectrally surjective [14,
Theorem 2.2]. This follows from Scholium A, Property (L).
Example 2.5. Quasi-Prüfer domains R with quotient fields K can be
characterized by R ⊆ K is quasi-Prüfer. The reader may consult [9,
Theorem 1.1] or [18]. In view of [2, Theorem 2.7], R is a quasi-Prüfer
domain if and only if Spec(R(X)) → Spec(R) is bijective.
We give here another example of quasi-Prüfer extension. An extension R ⊂ S is called a going-down pair if each of its subextensions has
the going-down property. For such a pair, R ⊆ T has incomparability
for each T ∈ [R, S], at each non-maximal prime ideal of R [3, Lemma
5.8](ii). Now let M be a maximal ideal of R, whose fiber is not void in
T . Then R ⊆ T is a going-down pair, and so is R/M ⊆ T /MT because
MT ∩ R = M. By [3, Corollary 5.6], the dimension of T /MT is ≤ 1.
Therefore, if R ⊂ S is a going-down pair, then R ⊂ S is quasi-Prüfer
if and only if dim(T /MT ) 6= 1 for each T ∈ [R, S] and M ∈ Max(R).
Also open-ring pairs R ⊂ S are quasi-Prüfer by [8, Proposition 2.13].
An i-pair is an extension R ⊆ S such that Spec(T ) → Spec(R) is
injective for each T ∈ [R, S], or equivalently if and only if R ⊆ S is
quasi-Prüfer and R ⊆ R is spectrally injective [38, Proposition 5.8].
These extensions appear frequently in the integral domains context.
Another examples are given by some extensions R ⊆ S, such that
Spec(S) = Spec(R) as sets, as we will see later.
3. Properties of quasi-Prüfer extensions
We now develop the machinery of quasi-Prüfer extensions.
Proposition 3.1. An extension R ⊂ S is (quasi-)Prüfer if and only if
RP ⊆ SP is (quasi-)Prüfer for any P ∈ Spec(R) (P ∈ MSupp(S/R)).
Proof. The proof is easy if we use the INC-pair property definition of
quasi-Prüfer extension (see also [6, Proposition 2.4]).

Proposition 3.2. Let R ⊆ S be a quasi-Prüfer extension and ϕ : S →
S ′ an integral ring morphism. Then ϕ(R) ⊆ S ′ is quasi-Prüfer and
S ′ = ϕ(S)ϕ(R), where ϕ(R) is the integral closure of ϕ(R) in S ′ .
Proof. It is enough to apply [26, Theorem 5.9] to the Prüfer extension
R ⊆ S and to use Definition 2.1.

This result applies with S ′ := S ⊗R R′ , where R → R′ is an integral
morphism. Therefore integrality ascends the quasi-Prüfer property.

QUASI-PRÜFER EXTENSIONS

9

We know that a composite of Prüfer extensions is Prüfer [26, Theorem 5.6, p.51]. The following Corollary 3.3 contains [7, Theorem 3].
Corollary 3.3. Let R ⊆ T ⊆ S be a tower of extensions. Then R ⊆ S
is quasi-Prüfer if and only if R ⊆ T and T ⊆ S are quasi-Prüfer. It
follows that R ⊆ T is quasi-Prüfer if and only if R ⊆ RT is quasiPrüfer.
Proof. Consider a tower (T ) of extensions R ⊆ R ⊆ S := R′ ⊆ R′ ⊆ S ′
(a composite of two quasi-Prüfer extensions). By using Proposition 3.2
we see that R ⊆ S = R′ ⊆ R′ is quasi-Prüfer. Then (T ) is obtained
by writing on the left an integral extension and on the right a Prüfer
extension. Therefore, (T ) is quasi-Prüfer. We prove the converse.
If R ⊆ T ⊆ S is a tower of extensions, then R ⊆ T and T ⊆ S
are INC-pairs whenever R ⊆ S is an INC-pair. The converse is then a
consequence of Theorem 2.3.
The last statement is [7, Corollary 4].

Using the above corollary, we can exhibit new examples of quasiPrüfer extensions. We recall that a ring R is called Laskerian if each of
its ideals is a finite intersection of primary ideals and a ring extension
R ⊂ S a Laskerian pair if each T ∈ [R, S] is a Laskerian ring. Then [42,
Proposition 2.1] shows that if R is an integral domain with quotient
field F 6= R and F ⊂ K is a field extension, then R ⊂ K is a Laskerian
pair if and only if K is algebraic over R and R (in K) is a Laskerian
Prüfer domain. It follows easily that R ⊂ K is quasi-Prüfer.
Next result generalizes [25, Proposition 1].
Corollary 3.4. An FMC extension R ⊂ S is quasi-Prüfer.
Proof. Because R ⊂ S is a composite of finitely many minimal extensions, by Corollary 3.3, it is enough to observe that a minimal extension
is either Prüfer or integral.

Corollary 3.5. Let R ⊆ S be a quasi-Prüfer extension and a tower
R ⊆ T ⊆ S, where R ⊆ T is integrally closed. Then R ⊆ T is Prüfer.
T

Proof. Observe that R ⊆ T is quasi-Prüfer and then that R = R . 
Next result deals with the Dobbs-Shapiro pseudo-Prüfer extensions
of integral domains [15], that they called pseudo-normal pairs. Suppose
that R is local, we call here pseudo-Prüfer an extension R ⊆ S such that
there exists T ∈ [R, S] with Spec(R) = Spec(T ) and T ⊆ S is Prüfer
[15, Corollary 2.5]. If R is arbitrary, the extension R ⊆ S is called
pseudo-Prüfer if RM ⊆ SM is pseudo-Prüfer for each M ∈ Max(R).
In view of the Corollary 3.3, it is enough to characterize quasi-Prüfer
extensions of the type R ⊆ T with Spec(R) = Spec(T ).

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G. PICAVET AND M. PICAVET

Corollary 3.6. Let R ⊆ T be an extension with Spec(R) = Spec(T )
and (R, M) local. Then R ⊆ T is quasi-Prüfer if and only if Spec(R) =
Spec(U) for each U ∈ [R, T ] and, if and only if R/M ⊆ T /M is an
algebraic field extension. In such a case, R ⊆ T is Prüfer-closed.
Proof. It follows from [1] that M ∈ Max(T ). Part of the proof is gotten
by observing that R ⊆ U is an INC extension if Spec(R) = Spec(U).
e is a spectrally
Another one is proved in [1, Corollary 3.26]. Now R ⊆ R
e
surjective flat epimorphism and then, by Scholium A, R = R.


Let R ⊆ S be an extension and I an ideal shared with R and S. It
is easy to show that R ⊆ S is quasi-Prüfer if and only if R/I ⊆ S/I is
quasi-Prüfer by using [26, Proposition 5.8] in the Prüfer case. We are
able to give a more general statement.
Lemma 3.7. Let R ⊆ S be a (quasi-)Prüfer extension and J an ideal
of S with I = J ∩ R. Then R/I ⊆ S/J is a (quasi-)Prüfer extension.
If R ⊆ S is Prüfer and N is a maximal ideal of S, then R/(N ∩ R) is
a valuation domain with quotient field S/N.
Proof. Assume first that R ⊆ S is Prüfer. We have J = IS by
Scholium A(6), because R ⊆ S is a flat epimorpism. Therefore, any
D ∈ [R/I, S/J] is of the form C/J where C ∈ [R, S]. We can write
C/IS = (C + I)/IS ∼
= C/C ∩ IS. As R ⊆ C is a flat epimorphism,
C ∩ IS = IC. It then follows that D = C ⊗ R/I and we get easily
that R/I ⊆ S/J is Prüfer, since R/I ⊆ D is a flat epimorphism. The
quasi-Prüfer case is an easy consequence.

With this lemma we generalize and complete [23, Proposition 1.1].
Proposition 3.8. Let R ⊆ S be an extension of rings. The following
statements are equivalent:
(1) R ⊆ S is quasi-Prüfer;
(2) R/(Q ∩ R) ⊆ S/Q is quasi-Prüfer for each Q ∈ Spec(S) ;
(3) (X − s)S[X] ∩ R[X] 6⊆ M[X] for each s ∈ S and M ∈ Max(R);
(4) For each T ∈ [R, S], the fiber morphisms of R ⊆ T are integral.
Proof. (1) ⇒ (2) is entailed by Lemma 3.7. Assume that (2) holds and
let M ∈ Max(R) that contains a minimal prime ideal P , lain over by a
minimal prime ideal Q of S. Then (2) ⇒ (3) follows from [23, Proposition 1.1(1)], applied to R/(Q ∩ R) ⊆ S/Q. If (3) holds, argue as in
the paragraph before [23, Proposition 1.1] to get that R ⊆ S is a Pextension, whence an INC-extension by [14, Proposition 2.1]. Because
integral extensions have incomparability, we see that (4) ⇒ (1). Corollary 3.3 shows that the reverse implication holds, if any quasi-Prüfer

QUASI-PRÜFER EXTENSIONS

11

extension R ⊆ S has integral fiber morphisms. For P ∈ Spec(R), the
extension RP /P RP ⊆ SP /P SP is quasi-Prüfer by Lemma 3.7. The
ring RP /P RP is zero-dimensional and RP /P RP → SP /P SP , being a
flat epimorphism, is therefore surjective by Scholium A (S). It follows
that the fiber morphism at P is integral.

Remark 3.9. The logical equivalence (1) ⇔ (2) is still valid if we
replace quasi-Prüfer with integral in the above proposition. It is enough
to show that an extension R ⊆ S is integral when R/P ⊆ S/Q is
integral for each Q ∈ Spec(S) and P := Q ∩ R. We can suppose that
S = R[s] ∼
= R[X]/I, where X is an indeterminate, I an ideal of R[X]
and Q varies in Min(S), because for an extension A ⊆ B, any element
of Min(A) is lain over by some element of Min(B). If Σ is the set of
unitary polynomials of R[X], the assumptions show that any element
of Spec(R[X]), containing I, meets Σ. As Σ is a multiplicatively closed
subset, I ∩ Σ 6= ∅, whence s is integral over R.
But a similar result does not hold if we replace quasi-Prüfer with
Prüfer, except if we suppose that R ⊆ S is integrally closed. To see
this, apply the above proposition to get a quasi-Prüfer extension R ⊆ S
if each R/P ⊆ S/Q is Prüfer. Actually, this situation already occurs
for Prüfer rings and their factor domains, as Lucas’s paper [29] shows.
More precisely, [29, Proposition 2.7] and the third paragraph of [29, p.
336] shows that if R is a ring with Tot(R) absolutely flat, then R is
a quasi-Prüfer ring if R/P is a Prüfer domain for each P ∈ Spec(R).
Now example [29, Example 2.4] shows that R is not necessarily Prüfer.
We observe that if R ⊆ S is quasi-Prüfer, then R/M is a quasiPrüfer domain for each N ∈ Max(S) and M := N ∩ R (in case R ⊆ S
is integral, R/M is a field). To prove this, observe that R/M ⊆ S/N
can be factored R/M ⊆ κ(M) ⊆ S/N. As we will see, R/M ⊆ κ(M)
is quasi-Prüfer because R/M ⊆ S/N is quasi-Prüfer.
The class of Prüfer extensions is not stable by (flat) base change.
For example, let V be a valuation domain with quotient field K. Then
V [X] ⊆ K[X] is not Prüfer [26, Example 5.12, p.53]. Thus if we
consider an ideal I of R and J := IS, R ⊆ S Prüfer may not imply
R/I ⊆ S/IS Prüfer except if IS ∩ R = I. This happens for instance
for a prime ideal I of R that is lain over by a prime ideal of S.
Proposition 3.10. Let R ⊆ S be a (quasi)-Prüfer extension and R →
T a flat epimorphism, then T ⊆ S ⊗R T is (quasi)-Prüfer. If in addition
S and T are both subrings of some ring and R ⊆ T is an extension,
then T ⊆ T S is (quasi)-Prüfer.

12

G. PICAVET AND M. PICAVET

Proof. For the first part, it is enough to consider the Prüfer case. It is
well known that the following diagram is a pushout if Q ∈ Spec(T ) is
lying over P in R:
RP −−−→


y

SP


y

TQ −−−→ (T ⊗R S)Q
As RP → TQ is an isomorphism since R → T is a flat epimorphism
by Scholium A, it follows that RP ⊆ SP identifies to TQ → (T ⊗R S)Q .
The result follows because Prüfer extensions localize and globalize.
In case R → T is a flat epimorphic extension, the surjective maps
T ⊗R S → T S and R ⊗R T → RT are isomorphisms because R → T R
(resp. S → ST ) is injective and R → T ⊗T R (resp. S → S ⊗R T ) is a
flat epimorphism. Then it is enough to use Scholium A.

The reader may find in [26, Corollary 5.11, p.53] that if R ⊆ A ⊆ S
and R ⊆ B ⊆ S are extensions and R ⊆ A and R ⊆ B are both Prüfer,
then R ⊆ AB is Prüfer.
Proposition 3.11. Let R ⊆ A and R ⊆ B be two extensions, where
A and B are subrings of a ring S. If they are both quasi-Prüfer, then
R ⊆ AB is quasi-Prüfer.
Proof. Let U and V be the integral closures of R in A and B. Then R ⊆
A ⊆ AV is quasi-Prüfer because A ⊆ AV is integral and Corollary 3.3
applies. Using again Corollary 3.3 with R ⊆ V ⊆ AV , we find that
V ⊆ AV is quasi-Prüfer. Now Proposition 3.10 entails that B ⊆ AB
is quasi-Prüfer because V ⊆ B is a flat epimorphism. Finally R ⊆ AB
is quasi-Prüfer, since a composite of quasi-Prüfer extensions.

It is known that an arbitrary product of extensions is Prüfer if and
only if each of its components is Prüfer [26, Proposition 5.20, p.56].
The following result is an easy consequence.
Proposition 3.12. Let {Ri ⊆ Si |i = 1, . . . , n} be a finite family of
quasi-Prüfer extensions, then R1 × · · · × Rn ⊆ S1 × · · · × Sn is quasiPrüfer. In particular, if {R ⊆ Si |i = 1, . . . , n} is a finite family of
quasi-Prüfer extensions, then R ⊆ S1 × · · · × Sn is quasi-Prüfer.
In the same way we have the following result deduced from [26,
Remark 5.14, p.54].
Proposition 3.13. Let R ⊆ S be an extension of rings and an upward
directed family {Rα |α ∈ I} of elements of [R, S] such that R ⊆ Rα is
quasi-Prüfer for each α ∈ I. Then R ⊆ ∪[Rα |α ∈ I] is quasi-Prüfer.

QUASI-PRÜFER EXTENSIONS

13

Proof. It is enough to use [26, Proposition 5.13, p.54] where Aα is the
integral closure of R in Rα .

A ring morphism R → T preserves the integral closure of ring morT ⊗R S
∼
phisms R → S if T
= T ⊗R R for every ring morphism R → S. An
absolutely flat morphism R → T (R → T and T ⊗R T → T are both
flat) preserves integral closure [33, Theorem 5.1]. Flat epimorphisms,
Henselizations and étale morphisms are absolutely flat. Another examples are morphisms R → T that are essentially of finite type and
(absolutely) reduced [37, Proposition 5.19](2). Such morphisms are flat
if R is reduced [28, Proposition 3.2].
We will prove an ascent result for absolutely flat ring morphisms.
This will be proved by using base changes. For this we need to introduce
some concepts. A ring A is called an AIC ring if each monic polynomial
of A[X] has a zero in A. We recalled in [35, p.4662] that any ring A
has a faithfully flat integral extension A → A∗ , where A∗ is an AIC
ring. Moreover, if A is an AIC ring, each localization AP at a prime
ideal P of A is a strict Henselian ring [35, Lemma II.2].
Theorem 3.14. Let R ⊆ S be a (quasi-) Prüfer extension and R → T
an absolutely flat ring morphism. Then T → T ⊗R S is a (quasi-)
Prüfer extension.
Proof. We can suppose that R is an AIC ring. To see this, it is enough
to use the base change R → R∗ . We set T ∗ := T ⊗R R∗ , S ∗ := S ⊗R R∗ .
We first observe that R∗ ⊆ S ∗ is quasi-Prüfer for the following reason:
the composite extension R ⊆ S ⊆ S ∗ is quasi-Prüfer because the last
extension is integral. Moreover, R∗ → T ∗ is absolutely flat. In case
T ∗ ⊆ T ∗ ⊗R∗ S ∗ is quasi-Prüfer, so is T ⊆ T ⊗R S, because T → T ∗ =
T ⊗R R∗ is faithfully flat and T ∗ ⊆ T ∗ ⊗R∗ S ∗ is deduced from T ⊆R S
by the faithfully flat base change T → T ⊗R S. It is then enough to
apply Proposition 3.17.
We thus assume from now on that R is an AIC ring.
Let N ∈ Spec(T ) be lying over M in R. Then RM → TN is absolutely
flat [32, Proposition f] and RM ⊆ SM is quasi-Prüfer. Now observe
that (T ⊗R S)N ∼
= TN ⊗RM SM . Therefore, we can suppose that R and
T are local and R → T is local and injective. We deduce from [33,
Theorem 5.2], that RM → TN is an isomorphism. Therefore the proof
is complete in the quasi-Prüfer case. For the Prüfer case, we need only
to observe that absolutely flat morphisms preserve integral closure and
a quasi-Prüfer extension is Prüfer if it is integrally closed.


14

G. PICAVET AND M. PICAVET

Proposition 3.15. Let R ⊆ S be an extension of rings and R → T a
base change which preserves integral closure. If T ⊆ T ⊗R S has FCP
and R ⊆ S is Prüfer, then T ⊆ T ⊗R S is Prüfer.
Proof. The result holds because an FCP extension is Prüfer if and only
if it is integrally closed.

e ⊆ Te needs not to be an isomorphism, since
We observe that T ⊗R R
this property may fail even for a localization R → RP , where P is a
prime ideal of R.
Proposition 3.16. Let R ⊆ S be an extension of rings, R → R′ a
faithfully flat ring morphism and set S ′ := R′ ⊗R S. If R′ ⊆ S ′ is
(quasi-) Prüfer (respectively, FCP), then so is R ⊆ S.
Proof. The Prüfer case is clear, because faithfully flat morphisms descend flat epimorphisms (Scholium A (9)). For the quasi-Prüfer case,
we use the INC-pair characterization and the fact that FR,S (P ) →
FR′ ,S ′ (P ′ ) is faithfully flat for P ′ ∈ Spec(R′ ) lying over P in R [22,
Corollaire 3.4.9]. The FCP case is proved in [11, Theorem 2.2].

Proposition 3.17. Let R ⊆ S be a ring extension and R → R′ a
spectrally surjective ring morphism (for example, either faithfully flat
or injective and integral). Then R ⊆ S is quasi-Prüfer if R′ → R′ ⊗R S
is injective (for example, if R → R′ is faithfully flat) and quasi-Prüfer.
Proof. Let T ∈ [R, S] and P ∈ Spec(R) and set T ′ := T ⊗R R′ . There
is some P ′ ∈ Spec(R′ ) lying over P , because R → R′ is spectrally
surjective. There is a faithfully flat morphism FR,T (P ) → FR′ ,T ′ (P ′) ∼
=
′
FR,T (P ) ⊗k(P ) κ(P ) [22, Corollaire 3.4.9]. By Theorem 2.3, the result

follows from the faithful flatness of FR,T (P ) → FR′ ,T ⊗R R′ (P ′ ).
Theorem 3.18. Let R ⊆ S be a ring extension.
=⇒
e
(1) R ⊆ S has a greatest quasi-Prüfer subextension R ⊆ R = R.
=⇒
e =: R
~ is quasi-Prüfer and then R
~ ⊆ R.
(2) R ⊆ RR
=⇒

(3) R

R

=⇒

e R = R.
e
= R and R

Proof. To see (1), use Proposition 3.13 which tells us that the set of all
quasi-Prüfer subextensions is upward directed and then use Proposi=⇒

=⇒

tion 3.12 to prove the existence of R . Then let R ⊆ T ⊆ R be a tower
=⇒
e ⊆ =⇒
R , we
with R ⊆ T integral and T ⊆ R Prüfer. From T ⊆ R ⊆ R

=⇒
e
deduce that T = R and then R = R.
e can be factored R ⊆ R
e ⊆ RR
e and is a tower of
(2) Now R ⊆ RR
e → RR
e is integral.
quasi-Prüfer extensions, because R

QUASI-PRÜFER EXTENSIONS

15
=⇒

(3) Clearly, the integral closure and the Prüfer closure of R in R are
=⇒
=⇒
e with R , and R, R
e ⊆ R.
the respective intersections of R and R


This last result means that, as long integral closures and Prüfer
=⇒

closures of subsets of R are concerned, we can suppose that R ⊆ S is
quasi-Prüfer.
4. Almost-Prüfer extensions
We next give a definition “dual” of the definition of a quasi-Prüfer
extension.
4.1. Arbitrary extensions.
Definition 4.1. A ring extension R ⊆ S is called an almost-Prüfer
extension if it can be factored R ⊆ T ⊆ S, where R ⊆ T is Prüfer and
T ⊆ S is integral.
Proposition 4.2. An extension R ⊆ S is almost-Prüfer if and only if
e ⊆ S is integral. It follows that the subring T of the above definition
R
e=R
b when R ⊆ S is almost-Prüfer.
is R
Proof. If R ⊆ S is almost-Prüfer, there is a factorization R ⊆ T ⊆
e⊆R
b ⊆ S, where T ⊆ R
b is both integral and a flat epimorphism by
R
e=R
b by Scholium A (5) (L).
Scholium A (4). Therefore, T = R

Corollary 4.3. Let R ⊆ S be a quasi-Prüfer extension, and let T ∈

^
∩T
[R, S]. Then, T ∩ R ⊆ T R is almost-Prüfer. Moreover, T = R

TR

.

Proof. T ∩ R ⊆ T is quasi-Prüfer by Corollary 3.3. Being integrally
closed, it is Prüfer by Corollary 3.5. Moreover, T ⊆ T R is an integral
^
extension. Then, T ∩ R ⊆ T R is almost-Prüfer and T = R
∩T

TR

. 

We note that integral extensions and Prüfer extensions are almostPrüfer and hence minimal extensions are almost-Prüfer. There are
quasi-Prüfer extensions that are not almost-Prüfer. It is enough to
consider [39, Example 3.5(1)]. Let R ⊆ T ⊆ S be two minimal extensions, where R is local, R ⊆ T integral and T ⊆ S is Prüfer. Then
b and
R ⊆ S is quasi-Prüfer but not almost-Prüfer, because S = R
e The same example shows that a composite of almost-Prüfer
R = R.
extensions may not be almost-Prüfer.
But the reverse implication holds.

16

G. PICAVET AND M. PICAVET

Theorem 4.4. Let R ⊆ S be an almost-Prüfer extension. Then R ⊆ S
e = R,
b (R)
e P =R
fP for each P ∈ Spec(R).
is quasi-Prüfer. Moreover, R
In this case, any flat epimorphic subextension R ⊆ T is Prüfer.
e ⊆ S, be an almost-Prüfer extension, that is R
e ⊆ S is
Proof. Let R ⊆ R
e is Prüfer. Now the Morita
integral. The result follows because R ⊆ R
hull and the Prüfer hull coincide by Proposition 4.2. In the same way,
e P →R
fP is a flat epimorphism and (R)
e P → SP is integral.
(R)


We could define almost-Prüfer rings as the rings R such that R ⊆
e = Tot(R) (by TheoTot(R) is almost-Prüfer. But in that case R
rem 4.4), so that R is a Prüfer ring. The converse evidently holds.
Therefore, this concept does not define something new.
We observed in [10, Remark 2.9(c)] that there is an almost-Prüfer
FMC extension R ⊆ S ⊆ T , where R ⊆ S is a Prüfer minimal extension and S ⊆ T is minimal and integral. But R ⊆ T is not an FCP
extension.
Proposition 4.5. Let R ⊆ S be an extension verifying the hypotheses:
(i) R ⊆ S is quasi-Prüfer.
(ii) R ⊆ S can be factored R ⊆ T ⊆ S, where R ⊆ T is a flat
epimorphism.
(1) Then the following commutative diagram (D) is a pushout,
R −−−→


y

R


y

T −−−→ T R

T R ⊆ S is Prüfer and R ⊆ T R is quasi-Prüfer. Moreover,
FR,R (P ) ∼
= FT,T R (Q) for each Q ∈ Spec(T ) and P := Q ∩ R.
(2) If in addition R ⊆ T is integrally closed, (D) is a pullback,
T ∩ R = R, (R : R) = (T : T R) ∩ R and (T : T R) = (R : R)T .
Proof. (1) Consider the injective composite map R → R ⊗R T → T R.
As R → R ⊗R T is a flat epimorphism, because deduced by a base
change of R → T , we get that the surjective map R ⊗R T → T R
is an isomorphism by Scholium A (3). By fibers transitivity, we have
FT,RT (Q) ∼
= κ(Q)⊗k(P ) FR,R (P ) [22, Corollaire 3.4.9]. As κ(P ) → κ(Q)
is an isomorphism by Scholium A, we get that FR,R (P ) ∼
= FT,RT (Q).
(2) As in [5, Lemma 3.5], R = T ∩ R. The first statement on the
conductors has the same proof as in [5, Lemma 3.5]. The second holds
because R ⊆ T is a flat epimorphism (see Scholium A (6)).


QUASI-PRÜFER EXTENSIONS

17

Theorem 4.6. Let R ⊂ S be a quasi-Prüfer extension and the diagram
(D’):
R −−−→ R




y
y
(1)
(2)
(3)
(4)
(5)

e −−−→ RR
e
R
e = R and
(D’) is a pushout and a pullback, such that R ∩ R
e : RR)
e ∩ R so that (R
e : RR)
e = (R : R)R.
e
(R : R) = (R
=⇒
e = S, where
e=R
~ ⊆ R =R
e =R
R ⊂ S can be factored R ⊆ RR
the first extension is almost-Prüfer and the second is Prüfer.
e = R.
e
e⇔R
R ⊂ S is almost-Prüfer if and only if S = RR
e =R
e=R
~ is the greatest almost-Prüfer subextension of
R ⊆ RR
e=R
eR~ .
R ⊆ S and R
e
Supp(S/R) = Supp(R/R)
∪ Supp(R/R) if R ⊆ S is almostPrüfer. (Supp can be replaced with MSupp).

Proof. To show (1), (2), in view of Theorem 3.18, it is enough to apply
=⇒
e and S = R , because R ⊆ RR
e is almostProposition 4.5 with T = R
Prüfer whence quasi-Prüfer, keeping in mind that a Prüfer extension is
integrally closed, whereas an integral Prüfer extension is trivial. Moree = RR
e because RR
e⊆R
e is both integral and integrally closed.
over, R
(3) is obvious.
(4) Now consider an almost-Prüfer subextension R ⊆ T ⊆ U, where
R ⊆ T is Prüfer and T ⊆ U is integral. Applying (3), we see that
U U
e ⊆ RR
e in view of Proposition 1.6.
U =R R
e
(5) Obviously, Supp(R/R)
∪ Supp(R/R) ⊆ Supp(S/R). Conversely,
e M = RM .
let M ∈ Spec(R) be such that RM 6= SM , and RM = (R)
e M = RM , which is absurd.
Then (3) entails that SM = (R)M (R)


Corollary 4.7. Let R ⊆ S be an almost-Prüfer extension. The following conditions are equivalent:
(1) Supp(S/R) ∩ Supp(R/R) = ∅.
e ∩ Supp(R/R)
e
(2) Supp(S/R)
= ∅.
e
(3) Supp(R/R) ∩ Supp(R/R) = ∅.

eP = R
fP for each
Proof. Since R ⊆ S is almost-Prüfer, we get (R)
e
P ∈ Spec(R). Moreover, Supp(S/R) = Supp(R/R) ∪ Supp(R/R) =
e ∪ Supp(R/R).
e
Supp(S/R) ∪ Supp(R/R) = Supp(S/R)
e ∩ Supp(R/R).
e
(1) ⇒ (2): Assume that there exists P ∈ Supp(S/R)
e P 6= SP , RP , so that RP ⊂ SP is neither Prüfer, nor integral.
Then, (R)

18

G. PICAVET AND M. PICAVET

But, P ∈ Supp(S/R) = Supp(S/R) ∪ Supp(R/R). If P ∈ Supp(S/R),
then P 6∈ Supp(R/R), so that (R)P = RP and RP ⊂ SP is Prüfer,
a contradiction. If P ∈ Supp(R/R), then P 6∈ Supp(S/R), so that
(R)P = SP and RP ⊂ SP is integral, a contradiction.
e
(2) ⇒ (3): Assume that there exists P ∈ Supp(R/R)
∩ Supp(R/R).
e
Then, RP 6= (R)P , (R)P , so that RP ⊂ SP is neither Prüfer, nor ine ∪ Supp(R/R).
e
tegral. But, P ∈ Supp(S/R) = Supp(S/R)
If P ∈
e
e
e
Supp(S/R), then P 6∈ Supp(R/R), so that (R)P = RP and RP ⊂ SP
e
e
is integral, a contradiction. If P ∈ Supp(R/R),
then P 6∈ Supp(S/R),
e P = SP and RP ⊂ SP is Prüfer, a contradiction.
so that (R)
(3) ⇒ (1): Assume that there exists P ∈ Supp(S/R) ∩ Supp(R/R).
Then, (R)P 6= RP , SP , so that RP ⊂ SP is neither Prüfer, nor integral.
e
e
If P ∈ Supp(R/R),
But, P ∈ Supp(S/R) = Supp(R/R) ∪ Supp(R/R).
then P 6∈ Supp(R/R), so that (R)P = RP and RP ⊂ SP is Prüfer,
e
a contradiction. If P ∈ Supp(R/R), then P 6∈ Supp(R/R),
so that
e
(R)P = RP and RP ⊂ SP is integral, a contradiction.

Proposition 4.5 has the following similar statement proved by Ayache
and Dobbs. It reduces to Theorem 4.6 in case R ⊆ S has FCP because
of Proposition 1.3.

Proposition 4.8. Let R ⊆ T ⊆ S be a quasi-Prüfer extension, where
T ⊆ S is an integral minimal extension and R ⊆ T is integrally closed .
Then the diagram (D) is a pullback, S = T R and (T : S) = (R : R)T .
Proof. [5, Lemma 3.5].

Proposition 4.9. Let R ⊆ U ⊆ S and R ⊆ V ⊆ S be two towers
of extensions, such that R ⊆ U and R ⊆ V are almost-Prüfer. Then
g=U
e Ve .
R ⊆ UV is almost-Prüfer and UV
Proof. Denote by U ′ , V ′ and W ′ the Prüfer hulls of R in U, V and
W = UV . We deduce from [26, Corollary 5.11, p.53], that R ⊆ U ′ V ′
is Prüfer. Moreover, U ′ V ′ ⊆ UV is clearly integral and U ′ V ′ ⊆ W ′
because the Prüfer hull is the greatest Prüfer subextension. We deduce
g=U
e Ve .
that R ⊆ UV is almost-Prüfer and that UV

Proposition 4.10. Let R ⊆ U ⊆ S and R ⊆ V ⊆ S be two towers
of extensions, such that R ⊆ U is almost-Prüfer and R ⊆ V is a flat
epimorphism. Then U ⊆ UV is almost-Prüfer.

Proof. Mimic the proof of Proposition 4.9 and use [26, Theorem 5.10,
p.53].

Proposition 4.11. Let R ⊆ S be an almost-Prüfer extension and
R → T a flat epimorphism. Then T ⊆ T ⊗R S is almost-Prüfer.

QUASI-PRÜFER EXTENSIONS

Proof. It is enough to use Proposition 3.10 and Definition 4.1.

19



Proposition 4.12. An extension R ⊆ S is almost-Prüfer if and only
fP = (R)
e P for each P ∈ Spec(R).
if RP ⊆ SP is almost-Prüfer and R
e P ⊆R
fP . Suppose
Proof. For an arbitrary extension R ⊆ S we have (R)
eP =R
fP by
that R ⊆ S is almost-Prüfer, then so is RP ⊆ SP and (R)
Theorem 4.4. Conversely, if R ⊆ S is locally almost-Prüfer, whence
fP = (R)
e P holds
locally quasi-Prüfer, then R ⊆ S is quasi-Prüfer. If R
e P so that S = RR
e and
for each P ∈ Spec(R), we have SP = (RR)
R ⊆ S is almost-Prüfer by Theorem 4.6.


Corollary 4.13. An FCP extension R ⊆ S is almost-Prüfer if and
only if RP ⊆ SP is almost-Prüfer for each P ∈ Spec(R).
Proof. It is enough to show that R ⊆ S is almost-Prüfer if RP ⊆ SP
is almost-Prüfer for each P ∈ Spec(R) using Proposition 4.12. Any
e ⊂ R1 is integral by definition of R.
e Assume that
minimal extension R
′
′
]
e P ⊂ (R
e
e
(R)
P ), so that there exists R2 ∈ [R, S] such that (R)P ⊂ (R2 )P
e P , for
is a Prüfer minimal extension with crucial maximal ideal Q(R)
e with Q ∩ R ⊆ P . In particular, R
e ⊂ R′ is not
some Q ∈ Max(R)
2
e R′ ] such that
integral. We may assume that there exists R1′ ∈ [R,
2
e Using
R1′ ⊂ R2′ is a Prüfer minimal extension with P 6∈ Supp(R1′ /R).
e R′ ] such that R
e ⊂ R2 is a
[39, Lemma 1.10], there exists R2 ∈ [R,
2
Prüfer minimal extension with crucial maximal ideal Q, a contradic]
e P ⊂ SP is integral for each P , whence (R)
e P = (R
tion. Then, (R)
P ). 
We now intend to demonstrate that our methods allow us to prove
easily some results. For instance, next statement generalizes [5, Corollary 4.5] and can be fruitful in algebraic number theory.

Proposition 4.14. Let (R, M) be a one-dimensional local ring and
R ⊆ S a quasi-Prüfer extension. Suppose that there is a tower R ⊂
T ⊆ S, where R ⊂ T is integrally closed. Then R ⊆ S is almost-Prüfer,
e and S is zero-dimensional.
T =R

Proof. Because R ⊂ T is quasi-Prüfer and integrally closed, it is Prüfer.
If some prime ideal of T is lying over M, R ⊂ T is a faithfully flat
epimorphism, whence an isomorphism by Scholium A, which is absurd.
Now let N be a prime ideal of T and P := N ∩ R. Then RP is zerodimensional and isomorphic to TN . Therefore, T is zero-dimensional.
It follows that T R is zero-dimensional. Since RT ⊆ S is Prüfer, we
deduce from Scholium A, that RT = S. The proof is now complete. 

20

G. PICAVET AND M. PICAVET

We also generalize [5, Proposition 5.2] as follows.
Proposition 4.15. Let R ⊂ S be a quasi-Prüfer
extension, such that
q
R is local with maximal ideal N := (R : R). Then R is local and
[R, S] = [R, R] ∪ [R, S]. If in addition R is one-dimensional, then
either R ⊂ S is integral or there is some minimal prime ideal P of R,
such that S = (R)P , P = SP and R/P is a one-dimensional valuation
domain with quotient field S/P .
Proof. R is obviously local. Let T ∈ [R, S] \ [R, R] and s ∈ T \ R.
Then s ∈ U(S) and s−1 ∈ R by Proposition 1.2 (1). But s−1 6∈ U(R),
so that s−1 ∈ N. It follows that there exists some integer n such that
s−n ∈ (R : R), giving s−n R ⊆ R, or, equivalently, R ⊆ Rsn ⊆ T .
Then, T ∈ [R, S] and we obtain [R, S] = [R, R] ∪ [R, S].
Assume that R is one-dimensional. If R ⊂ S is not integral then
R ⊂ S is Prüfer and R is one-dimensional. To complete the proof, use
Proposition 1.2 (3).

4.2. FCP extensions. In case we consider only FCP extensions, we
obtain more results.
Proposition 4.16. Let R ⊆ S be an FCP extension. The following
statements are equivalent:
(1) R ⊆ S is almost-Prüfer.
(2) RP ⊆ SP is either integral or Prüfer for each P ∈ Spec(R).
e ∩ Supp(R/R)
e
(3) RP ⊆ SP is almost-Prüfer and Supp(S/R)
= ∅.
(4) Supp(R/R) ∩ Supp(S/R) = ∅.
Proof. The equivalence of Proposition 4.12 shows that (2) ⇔ (1) holds
because Tb = Te and over a local ring T , an almost-Prüfer FCP extension
T ⊆ U is either integral or Prüfer [39, Proposition 2.4] . Moreover when
e P =R
fP
RP ⊆ SP is either integral or Prüfer, it is easy to show that (R)
Next we show that (3) is equivalent to (2) of Proposition 4.12.
e ∩ Supp(R/R)
e
Let P ∈ Supp(S/R)
be such that RP ⊆ SP is almoste
e P ⊂ SP . Since R ⊂ R
e
Prüfer. Then, (R)P 6= RP , SP , so that RP ⊂ (R)
e P , giving (R)
e P ⊆R
fP and RP 6= R
fP . It follows
is Prüfer, so is RP ⊂ (R)
fP = SP in view of the dichotomy principle [39, Proposition 3.3]
that R
fP 6= (R)
e P.
since RP is a local ring, and then R
fP 6= (R)
e P , i.e. P ∈ Supp(S/R). Then,
Conversely, assume that R
fP , so that R
fP = SP , as we have just seen. Hence RP ⊂ SP
RP 6= R
is integrally closed. It follows that RP = RP = RP , so that P 6∈
e
eP 6=
by Theorem 4.6(5). Moreover, R
Supp(R/R) and P ∈ Supp(R/R)

QUASI-PRÜFER EXTENSIONS

21

e
e ∩
SP implies that P ∈ Supp(S/R).
To conclude, P ∈ Supp(S/R)
e
Supp(R/R).
(1) ⇔ (4) An FCP extension is quasi-Prüfer by Corollary 3.4. Supe we
pose that R ⊆ S is almost-Prüfer. By Theorem 4.6, letting U := R,
get that U ∩R = R and S = RU. We deduce from [39, Proposition 3.6]
that Supp(R/R) ∩ Supp(S/R) = ∅. Suppose that this last condition
holds. Then by [39, Proposition 3.6] R ⊆ S can be factored R ⊆ U ⊆ S,
where R ⊆ U is integrally closed, whence Prüfer by Proposition 1.3,
and U ⊆ S is integral. Therefore, R ⊆ S is almost-Prüfer.

Lemma 4.17. Let B ⊂ D and C ⊂ D be two integral minimal extensions and A := B ∩ C. If A ⊂ D has FCP, then, A ⊂ D is integral.
Proof. Set M := (B : D) and N := (C : D).
If M 6= N, then, A ⊂ D is integral by [13, Proposition 6.6].
Assume that M = N. Then, M ∈ Max(A) by [13, Proposition 5.7].
Let B ′ be the integral closure of A in B. Then M is also an ideal of
B ′ , which is prime in B ′ , and then maximal in B ′ . If A ⊂ D is an
FCP extension, so is B ′ ⊆ B, which is a flat epimorphism, and so is
B ′ /M ⊆ B/M. Then, B ′ = B since B ′ /M is a field. It follows that
A ⊆ B is an integral extension, and so is A ⊂ D.

Proposition 4.18. Let R ⊂ S be an FCP almost-Prüfer extension.
e=R
b is the least T ∈ [R, S] such that T ⊆ S is integral.
Then, R

Proof. We may assume that R ⊂ S is not integral. If there is some
e such that U ⊆ R
e is integral, then U = R.
e Set X := {T ∈
U ∈ [R, R]
e is a minimal element of X.
[R, S] | T ⊆ S integral}. It follows that R
e is the least element of X.
We are going to show that R
e S] ≥ 1 and let R
e = R0 ⊂ R1 ⊂ · · · ⊂ Rn−1 ⊂ Rn = S
Set n := ℓ[R,
e S], with length n. There does not exist a
be a maximal chain of [R,
e
maximal chain of R-subalgebras
of S with length > n. Let T ∈ X. We
e
intend to show that T ∈ [R, S]. It is enough to choose T such that T
is a minimal element of X. Consider the induction hypothesis: (Hn ):
e S] when n := ℓ[R,
e S].
X ⊆ [R,
e ⊂ S is minimal. Let T ∈ X and
We first show (H1 ). If n = 1, R
e
T1 ∈ [T, S] be such that T1 ⊂ S is minimal. Assume that T1 6= R.
e⊂R
e is integral, which contradicts the
Lemma 4.17 shows that T1 ∩ R
e so that T = R
e for the same
beginning of the proof. Then, T1 = R,
contradiction and (H1 ) is proved.
Assume that n > 1 and that (Hk ) holds for any k < n. Let T ∈ X
e S], then
and T1 ∈ [T, S] be such that T1 ⊂ S is minimal. If T1 ∈ [R,

22

G. PICAVET AND M. PICAVET

e T1 ] ≤ n−1. But we get that T ∈ [R, T1 ], with T ⊆ T1 integral.
k := ℓ[R,
e is also the Prüfer hull of R ⊆ T1 , with k := ℓ[R,
e T1 ] ≤ n−1.
Moreover, R
e T1 ] ⊂ [R,
e S].
Since (Hk ) holds, we get that T ∈ [R,
e
If T1 6∈ [R, S], set U := T1 ∩Rn−1 . We get that T1 ⊂ S and Rn−1 ⊂ S
are minimal and integral. Using again Lemma 4.17, we get that U ⊂ S
e Rn−1 ] = n − 1 and U ∈ [R, Rn−1 ]. As before, R
e is
is integral, with ℓ[R,
e Rn−1 ],
also the Prüfer hull of R ⊆ Rn−1 . Since (Hn−1 ) holds, U ∈ [R,
e
so that T1 ∈ [R, S], a contradiction. Therefore, (Hn ) is proved.


We will need a relative version of the support. Let f : R → T be a
ring morphism and E a T -module. The relative support of E over R is
SR (E) := a f (SuppT (E)) and MSR (E) := SR (E)∩Max(R). In particular, for a ring extension R ⊂ S, we have SR (S/R) := SuppR (S/R)).
Proposition 4.19. Let R ⊆ S be an FCP extension. The following
statements hold:
e
(1) Supp(R/R)
∩ Supp(R/R) = ∅.
e
e R)
e ∩ Supp(R/R)
e
(2) Supp(R/R)
∩ Supp(R/R) = Supp(R/
= ∅.
e
(3) MSupp(S/R) = MSupp(R/R) ∪ MSupp(R/R).

e is
Proof. (1) is a consequence of Proposition 4.16(4) because R ⊆ R
almost-Prüfer.
e
We prove the first part of (2). If some M ∈ Supp(R/R)∩Supp(R/R),
′
e M , T := (R)M
it can be supposed in Max(R). Set R := RM , U := (R)
and M ′ := MRM . Then, R′ 6= U, T , with R′ ⊂ U FCP Prüfer and
R′ ⊂ T FCP integral, an absurdity [39, Proposition 3.3].
e R)
e ∩
To show the second part, assume that some P ∈ Supp(R/
e
Supp(R/R).
Then, P 6∈ Supp(R/R) by the first part of (2), so that

e P = RP R
eP = R
eP , a contradiction.
RP = RP , giving (R)
S
S
(3) Obviously, MSupp(S/R) = MS (S/R) = MS (S/T )∪MS (T /T )
T
T
∪MS (T /U )∪MS (U /U)∪MS (U/R). By [39, Propositions 2.3 and
S
S
T
3.2], we have MS (S/T ) ⊆ S (T /T ) = S (R/R ) = MS (R/R) =
T
T
MSupp(R/R), MS (T /U ) = S (R /R) ⊆ S (R/R) = Supp(R/R)
T
T
and MS (U /U) = S (R /R) = Supp(R/R). To conclude, MSupp(S/R) =
e
MSupp(R/R)
∪ MSupp(R/R).


Proposition 4.20. Let R ⊂ S be an FCP extension and M ∈ MSupp(S/R),
g
e
e
e
then R
M = (R)M if and only if M 6∈ MSupp(S/R) ∩ MSupp(R/R).
g
e
Proof. In fact, we are going to show that R
M 6= (R)M if and only if
e ∩ MSupp(R/R).
e
M ∈ MSupp(S/R)

QUASI-PRÜFER EXTENSIONS

23

e ∩ MSupp(R/R).
e
g
Let M ∈ MSupp(S/R)
Then, R
M 6= RM , SM and
g
e
g
then RM ⊂ R
⊂
S
.
Since
R
⊂
R
is
Prüfer,
so is RM ⊂ R
M
M
M
e M ⊆ R
g
g
by Proposition 1.2, giving (R)
and
R
=
6
R
.
Therefore,
M
M
M
g
g
e
R
=
S
[39,
Proposition
3.3]
since
R
is
local,
and
then
R
M
M
M
M 6= (R)M .
g
e
g
g
Conversely, if R
M 6= (R)M , then, RM 6= RM , so that RM = SM , as
we have just seen and then RM ⊂ SM is integrally closed. It follows
that RM = RM = RM , so that M 6∈ MSupp(R/R). Hence, M ∈
e
eM 6= SM ⇒ M ∈
MSupp(R/R)
by Proposition 4.19(3). Moreover, R
e To conclude, M ∈ MSupp(S/R)
e ∩ MSupp(R/R).
e
MSupp(S/R).

g
e
If R ⊆ S is any ring extension, with dim(R) = 0, then R
M = (R)M
for any M ∈ Max(R). Indeed by Scholium A (2), the flat epimorphism
e is bijective as well as RM → (R)
e M . This conclusion is still
R → R
valid in another context.

Corollary 4.21. Let R ⊂ S be an FCP extension. Assume that one
of the following conditions is satisfied:
e ∩ MSupp(R/R)
e
(1) MSupp(S/R)
= ∅.
e
(2) S = RR, or equivalently, R ⊆ S is almost-Prüfer.
g
e
Then, R
M = (R)M for any M ∈ Max(R).

Proof. (1) is Proposition 4.20. (2) is Proposition 4.12.



Proposition 4.22. Let R ⊂ S be an almost-Prüfer FCP extension.
e in T R.
e
Then, any T ∈ [R, S] is the integral closure of T ∩ R

e and V := T R.
e Since R ⊂ S is almost-Prüfer,
Proof. Set U := T ∩ R
e
e
e is also the Prüfer hull of
U ⊆ R is Prüfer and R ⊆ V is integral and R
U ⊆ V . Because R ⊂ S is almost-Prüfer, for each M ∈ MSuppR (S/R),
RM ⊆ SM is either integral, or Prüfer by Proposition 4.16, and so is
g
e
UM ⊆ VM . But R
M = (R)M by Corollary 4.21 is also the Prüfer hull
′
′
of UM ⊆ VM . Let T be the integral closure of U in V . Then, TM
is
the integral closure of UM in VM .
′
e M,
Assume that UM ⊆ VM is integral. Then VM = TM
and UM = (R)
e M = TM , giving TM = T ′ .
so that VM = TM (R)
M
′
e M,
Assume that UM ⊆ VM is Prüfer. Then UM = TM
and VM = (R)
′
e M = TM , giving TM = T .
so that UM = TM ∩ (R)
M
′
To conclude, we get that TM = TM
for each M ∈ MSuppR (S/R).
′
Since RM = SM , with TM = TM
for each M ∈ Max(R)\MSuppR (S/R),
′
we get T = T , whence T is the integral closure of U ⊆ V .


24

G. PICAVET AND M. PICAVET

We build an example of an FCP extension R ⊂ S where we have
g
e M for some M ∈ Max(R). In particular, R ⊂ S is not
RM 6= (R)
almost-Prüfer.
Example 4.23. Let R be an integral domain with quotient field S
and Spec(R) := {M1 , M2 , P, 0}, where M1 6= M2 are two maximal
ideals and P a prime ideal satisfying P ⊂ M1 ∩ M2 . Assume that
there are R1 , R2 and R3 such that R ⊂ R1 is Prüfer minimal, with
C (R, R1 ) = M1 , R ⊂ R2 is integral minimal, with C (R, R2 ) = M2
and R2 ⊂ R3 is Prüfer minimal, with C (R2 , R3 ) = M3 ∈ Max(R2 )
such that M3 ∩ R = M2 and M2 R3 = R3 . This last condition is
satisfied when R ⊂ R2 is either ramified or inert. Indeed, in both
cases, M3 R3 = R3 ; moreover, in the ramified case, we have M32 ⊆ M2
and in the inert case, M3 = M2 [36, Theorem 3.3]. We apply [13,
Proposition 7.10] and [10, Lemma 2.4] several times. Set R2′ := R1 R2 .
Then, R1 ⊂ R2′ is integral minimal, with C (R1 , R2′ ) =: M2′ = M2 R1
and R2 ⊂ R2′ is Prüfer minimal, with C (R2 , R2′ ) =: M1′ = M1 R2 ∈
Max(R2 ). Moreover, M1′ 6= M3 , Spec(R1 ) = {M2′ , P1 , 0}, where P1
is the only prime ideal of R1 lying over P . But, P = (R : R1 ) by
[17, Proposition 3.3], so that P = P1 . Set R3′ := R3 R2′ . Then, R2′ ⊂
R3′ is Prüfer minimal, with C (R2′ , R3′ ) =: M3′ = M3 R2′ ∈ Max(R2′ )
and R3 ⊂ R3′ is Prüfer minimal, with C (R3 , R3′ ) = M1′′ = M1 R3 ∈
Max(R3 ). It follows that we have Spec(R3′ ) = {P ′ , 0} where P ′ is the
only prime ideal of R3′ lying over P . To end, assume that R3′ ⊂ S is
Prüfer minimal, with C (R3′ , S) = P ′. Hence, R2 is the integral closure
of R in S. In particular, R ⊂ S has FCP [10, Theorems 6.3 and
3.13] and is quasi-Prüfer. Since R ⊂ R1 is integrally closed, we have
e Assume that R1 6= R.
e Then, there exists T ∈ [R1 , S] such
R1 ⊆ R.
that R1 ⊂ T is Prüfer minimal and C (R1 , T ) = M2′ , a contradiction
by Proposition 4.16 since M2′ = C (R1 , R2′ ), with R1 ⊂ R2′ integral
e It follows that M1 ∈ MSupp(R/R).
e
minimal. Then, R1 = R.
But,
′
e
e
P = C (R3 , S) ∩ R ∈ Supp(S/R) and P ⊂ M1 give M1 ∈ MSupp(S/R),
g
e
so that R
M1 6= (R)M1 by Proposition 4.20 giving that R ⊂ S is not
almost-Prüfer.
We now intend to refine Theorem 4.6, following the scheme used in
[4, Proposition 4] for extensions of integral domains.
Proposition 4.24. Let R ⊆ S and U, T ∈ [R, S] be such that R ⊆ U is
integral and R ⊆ T is Prüfer. Then U ⊆ UT is Prüfer in the following
cases and R ⊆ UT is almost-Prüfer.
e
(1) Supp(R/R)∩Supp(R/R)
= ∅ (for example, if R ⊆ S has FCP).
(2) R ⊆ U preserves integral closure.

QUASI-PRÜFER EXTENSIONS

25

Proof. (1) We have ∅ = MSupp(U/R)∩MSupp(T /R), since U ⊆ R and
e Let M ∈ MSupp((UT )/R). For M ∈ MSupp(U/R), we have
T ⊆ R.
RM = TM and (UT )M = UM . If M ∈
/ MSupp(U/R), then UM = RM
and (UT )M = TM , so that UM ⊆ (UT )M identifies to RM ⊆ TM .
Let N ∈ Max(U) and set M := N ∩ R ∈ Max(R) since R ⊆ U is
integral. If M 6∈ Supp(R/R), then RM = RM = UM and N is the
only maximal ideal of U lying over M. It follows that UM = UN and
(UT )M = (UT )N by [10, Lemma 2.4]. Then, UN ⊆ (UT )N identifies
e
to RM ⊆ TM which is Prüfer. If M 6∈ Supp(R/R),
then RM = TM
gives UM = (UT )M , so that UN = (UT )N by localizing the precedent
equality and UN ⊆ (UT )N is still Prüfer. Therefore, U ⊆ UT is locally
Prüfer, whence Prüfer by Proposition 1.1.
(2) The usual reasoning shows that U ⊗R T ∼
= UT , so that U ⊆ UT
UT
UT
is integrally closed. Since U is contained in R , we get that U = R .
Now observe that R ⊆ UT is almost-Prüfer, whence quasi-Prüfer. It
follows that U ⊆ UT is Prüfer.

Next propositions generalize Ayache’s results of [4, Proposition 11].
Proposition 4.25. Let R ⊆ S be a quasi-Prüfer extension, T, T ′ ∈
[R, S] and U := T ∩ T ′ . The following statements hold:
Te = (T^
∩ R) for each T ∈ [R, S].
′
e
e
T ∩ T ⊆ T^
∩ T ′.
Let Supp(T /T ) ∩ Supp(Te/T ) = ∅ (this assumption holds if R ⊆
S has FCP). Then, T ⊆ T ′ ⇒ Te ⊆ Te′ .
e
(4) If Supp(U /U) ∩ Supp(U/U)
= ∅, then Te ∩ Te′ = T^
∩ T ′.

(1)
(2)
(3)

Proof. (1) We observe that R ⊆ T is quasi-Prüfer by Corollary 3.3.
Since T ∩ R is the integral closure of R in T , we get that T ∩ R ⊆ T is
∩ R.
Prüfer. It follows that T ∩ R ⊆ Te is Prüfer. We thus have Te ⊆ T^
e
To prove the reverse inclusion, we set V := T ∩ R and W := V ∩ T .
We have W ∩ R = Ve ∩ R = V , because V ⊆ Ve ∩ R is integral and
Prüfer since we have a tower V ⊆ Ve ∩ R ⊆ Ve . Therefore, V ⊆ W
is Prüfer because W ∈ [V, Ve ]. Moreover, T ⊆ Te ⊆ Ve , since V ⊆ Te
is Prüfer. Then, T ⊆ W is integral because W ∈ [T, T ], and we have
V ⊆ T ⊆ W . This entails that T = W = Ve ∩ T , so that T ⊆ Ve is
Prüfer. It follows that Ve ⊆ Te since T ∈ [V, Ve ].
(2) A quasi-Prüfer extension is Prüfer if and only if it is integrally
closed. We observe that T ∩ T ′ ⊆ Te ∩ Te′ is integrally closed, whence
Prüfer. It follows that Te ∩ Te′ ⊆ T^
∩ T ′.

26

G. PICAVET AND M. PICAVET

(3) Set U = T ∩ R and U ′ = T ′ ∩ R, so that U, U ′ ∈ [R, R] with
U ⊆ U ′ . In view of (1), we thus can suppose that T, T ′ ∈ [R, R]. It
follows that T ⊆ T ′ is integral and T ⊆ Te is Prüfer. We deduce from
Proposition 4.24(1) that T ′ ⊆ T ′ Te is Prüfer, so that TeT ′ ⊆ Te′ , because
Supp(T /T ) ∩ Supp(Te/T ) = ∅ and T = R. Therefore, we have Te ⊆ Te′ .
e /U) = ∅. Then, T ∩T ′ ⊂ T, T ′
(4) Assume that Supp(U /U)∩Supp(U
gives T^
∩ T ′ ⊆ Te ∩ Te′ in view of (3), so that T^
∩ T ′ = Te ∩ Te′ by (2). 

Proposition 4.26. Let R ⊆ S be a quasi-Prüfer extension and T ⊆ T ′
a subextension of R ⊆ S. Set U := T ∩ R, U ′ := T ′ ∩ R, V := T R and
V ′ := T ′ R. The following statements hold:
(1) T ⊆ T ′ is integral if and only if V = V ′ .
(2) T ⊆ T ′ is Prüfer if and only if U = U ′ .
(3) Assume that U ⊂ U ′ is integral minimal and V = V ′ . Then,
T ⊂ T ′ is integral minimal, of the same type as U ⊂ U ′ .
(4) Assume that V ⊂ V ′ is Prüfer minimal and U = U ′ . Then,
T ⊂ T ′ is Prüfer minimal.
(5) Assume that T ⊂ T ′ is minimal and set P := C(T, T ′ ).
(a) If T ⊂ T ′ is integral, then U ⊂ U ′ is integral minimal if and
only if P ∩ U ∈ Max(U).
(b) If T ⊂ T ′ is Prüfer, then V ⊂ V ′ is Prüfer minimal if and
only if there is exactly one prime ideal in V lying over P .

Proof. In [R, S] we have the integral extensions U ⊆ U ′ , T ⊆ V, T ′ ⊆
V ′ and the Prüfer extensions V ⊆ V ′ , U ⊆ T, U ′ ⊆ T ′ . Moreover, R
is also the integral closure of U ⊆ V ′ .
(1) is gotten by considering the extension T ⊆ V ′ , which is both
T ⊆ V ⊆ V ′ and T ⊆ T ′ ⊆ V ′ .
(2) is gotten by considering the extension U ⊆ T ′ , which is both
U ⊆ T ⊆ T ′ and U ⊆ U ′ ⊆ T ′ .
(3) Assume that U ⊂ U ′ is integral minimal and V = V ′ . Then,
T ⊂ T ′ is integral by (1) and T 6= T ′ because of (2). Set M := (U :
U ′ ) ∈ SuppU (U ′ /U). For any M ′ ∈ Max(U) such that M ′ 6= M, we
′
′
′
have UM ′ = UM
′ , so that TM ′ = T M ′ because UM ′ ⊆ TM ′ is Prüfer.
′
′
′
But, U ⊆ T is almost-Prüfer, giving T = T U . By Theorem 4.6,
(T : T ′ ) = (U : U ′ )T = MT 6= T because T 6= T ′ . We get that U ⊆ T
Prüfer implies that M 6∈ SuppU (T /U) and UM = TM . It follows that
T ′ M = TM U ′ M = U ′ M . Therefore, TM ⊆ T ′ M identifies to UM ⊆ U ′ M ,
which is minimal of the same type as U ⊂ U ′ by [13, Proposition 4.6].
Then, T ⊂ T ′ is integral minimal, of the same type as U ⊂ U ′ .
(4) Assume that V ⊂ V ′ is Prüfer minimal and U = U ′ . Then,
T ⊂ T ′ is Prüfer by (2) and T 6= T ′ because of (1). Set Q := C(V, V ′ )

QUASI-PRÜFER EXTENSIONS

27

and P := Q ∩ T ∈ Max(T ) since Q ∈ Max(V ). For any P ′ ∈ Max(T )
such that P ′ 6= P , and Q′ ∈ Max(V ) lying above P ′, we have VQ′ = VQ′ ′ ,
so that VP ′ = V ′ P ′ . It follows that T ′ P ′ ⊆ V ′ P ′ is integral, so that
TP ′ = T ′ P ′ and P ′ 6∈ SuppT (T ′/T ). We get that T ⊂ T ′ is Prüfer
minimal in view of [10, Proposition 6.12].
(5) Assume that T ⊂ T ′ is a minimal extension and set P := C(T, T ′ ).
(a) Assume that T ⊂ T ′ is integral. Then, V = V ′ and U 6= U ′
by (1) and (2). We can use Proposition 4.5 getting that P = (U :
U ′ )T ∈ Max(T ) and Q := (U : U ′ ) = P ∩ U ∈ Spec(U). It follows that
Q 6∈ SuppU (T /U), so that UQ = TQ and UQ′ = TQ′ . Then, UQ ⊂ UQ′ is
integral minimal, with Q ∈ SuppU (U ′ /U).
If Q 6∈ Max(U), then U ⊂ U ′ is not minimal by the properties of the
crucial maximal ideal.
Assume that Q ∈ Max(U) and let M ∈ Max(U), with M 6= Q.
′
Then, UM = UM
because M + Q = U, so that U ⊂ U ′ is a minimal
extension and (a) is gotten.
(b) Assume that T ⊂ T ′ is Prüfer. Then, V 6= V ′ and U = U ′ by (1)
and (2). Moreover, P T ′ = T ′ gives P V ′ = V ′ . Let Q ∈ Max(V ) lying
over P . Then, QV ′ = V ′ gives that Q ∈ SuppV (V ′ /V ). Moreover, we
have V ′ = V T ′ . Let P ′ ∈ Max(T ), P ′ 6= P . Then, TP ′ = TP′ ′ gives
VP ′ = VP′ ′ . It follows that SuppT (V ′ /V ) = {P } and SuppV (V ′ /V ) =
{Q ∈ Max(V ) | Q ∩ T = P }. But, by [10, Proposition 6.12], V ⊂ V ′ is
Prüfer minimal if and only if |SuppV (V ′ /V )| = 1, and then if and only
if there is exactly one prime ideal in V lying over P .

Lemma 4.27. Let R ⊆ S be an FCP almost-Prüfer extension and
U ∈ [R, R], V ∈ [R, S]. Then U ⊆ V has FCP and is almost-Prüfer.
Proof. Obviously, U ⊆ V has FCP and R is the integral closure of U
in V . Proposition 4.16 entails that SuppR (R/R) ∩ SuppR (S/R) = ∅.
We claim that SuppU (R/U) ∩ SuppU (V /R) = ∅. Deny and let Q ∈
SuppU (R/U) ∩ SuppU (V /R). Then, RQ 6= UQ , VQ . If P := Q ∩ R. we
get that RP 6= UP , VP , giving RP 6= RP , SP , a contradiction. Another
use of Proposition 4.16 shows that U ⊆ V is almost-Prüfer.

Proposition 4.28. Let R ⊆ S be an FCP almost-Prüfer extension and
T ⊆ T ′ a subextension of R ⊆ S. Set U := T ∩ R and V ′ := T ′ R. Let
W be the Prüfer hull of U ⊆ V ′ . Then, W is also the Prüfer hull of
T ⊆ T ′ and T ⊆ T ′ is an FCP almost-Prüfer extension.
Proof. By Lemma 4.27, we get that U ⊆ V ′ is an FCP almost-Prüfer
extension. Let Te be the Prüfer hull of T ⊆ T ′ . Since U ⊆ T and T ⊆ Te
are Prüfer, so is U ⊆ Te and Te ⊆ V ′ gives that Te ⊆ W . Then, T ⊆ W
is Prüfer as a subextension of U ⊆ W .

28

G. PICAVET AND M. PICAVET

Moreover, in view of Proposition 4.18, W is the least U-subalgebra
of V ′ over which V ′ is integral. Since T ′ ⊆ V ′ is integral, we get that
W ⊆ T ′ , so that W ∈ [T, T ′ ], with W ⊆ T ′ integral as a subextension
of W ⊆ V ′ . It follows that W is also the Prüfer hull of T ⊆ T ′ and
T ⊆ T ′ is an FCP almost-Prüfer extension.

5. The case of Nagata extensions
In this section we transfer the quasi-Prüfer (and almost-Prüfer) properties to Nagata extensions.
Proposition 5.1. Let R ⊆ S be a Prüfer (and FCP) extension, then
R(X) ⊆ S(X) is a Prüfer (and FCP) extension.
Proof. We can suppose that (R, M) is local, in order to use Proposition 1.2(3). Then it is enough to know the following facts: V (X) is
a valuation domain if so is V ; R[X]P [X] ∼
= RP (X) where
= R(X)P (X) ∼
∼
P (X) = P R(X) and R(X)/P (X) = (R/P )(X) for P ∈ Spec(R). If in
addition R ⊆ S is FCP, it is enough to use [11, Theorem 3.9]: R ⊂ S
has FCP if and only if R(X) ⊂ S(X) has FCP.

Proposition 5.2. If R ⊆ S is quasi-Prüfer, then so is R(X) ⊆ S(X),
R(X) = R(X) ∼
= R ⊗R R(X) and S(X) ∼
= S ⊗R R(X).
Proof. It is enough to use proposition 5.1, because R(X) = R(X). The
third assertion results from [34, Proposition 4 and Proposition 7]. 
Proposition 5.3. If R ⊆ S is almost-Prüfer, then so is R(X) ⊆ S(X).
^ = R(X)
e
It follows that R(X)
for an almost-Prüfer extension R ⊆ S.
e is Prüfer and R
e ⊆ S is
Proof. If R ⊆ S is almost-Prüfer, then R ⊆ R
e
e
integral and then R(X) ⊆ R(X)
is Prüfer and R(X)
⊆ S(X) is integral,
^ = R(X).
e
whence R(X) ⊆ S(X) is almost-Prüfer with R(X)

e = R.
Lemma 5.4. Let R ⊂ S be an FCP ring extension such that R
^ = R(X).
Then, R(X)

^ there is some T ′ ∈ [R(X), R(X)]
^ such that
Proof. If R(X) 6= R(X),
′
′
R(X) ⊂ T is Prüfer minimal. Set C (R(X), T ) ∈ MSupp(S(X)/R(X))
=: M ′ . There is M ∈ MSupp(S/R) such that M ′ = MR(X) [11,
Lemma 3.3]. But, M ′ 6∈ MSupp(R(X)/R(X)) = MSupp(R(X)/R(X))
by Proposition 4.19(2), giving that M 6∈ MSupp(R/R) = S (R/R).
Then [39, Proposition 1.7(3)] entails that M ∈ S (S/R). By [39,
Proposition 1.7(4)], there are some T1 , T2 ∈ [R, S] with T1 ⊂ T2 Prüfer
minimal (an FCP extension is quasi-Prüfer), with M = C (T1 , T2 ) ∩ R.

QUASI-PRÜFER EXTENSIONS

29

We can choose for T1 ⊂ T2 the first minimal extension verifying the preceding property. Therefore, M 6∈ S (T1 /R), so that M 6∈ S (T1 /R) =
Supp(T1 /R). By [39, Lemma 1.10], we get that there exists T ∈ [R, T2 ]
such that R ⊂ T is Prüfer minimal, a contradiction.

^
e
Proposition 5.5. If R ⊂ S is an FCP extension, then, R(X)
= R(X).
e is Prüfer, R(X) ⊆ R(X)
e
Proof. Because R ⊆ R
is Prüfer by Corol^ Assume that R(X)
^ and set
e
e
lary 5.1. Then, R(X)
⊆ R(X).
6= R(X)
^
e so that T = Te, giving T
e
T := R,
(X) = T (X) = R(X)
by Lemma 5.4.
^
^ is a Prüfer extension, contradicting the definition
Hence T
(X) ⊂ R(X)
^
^
e
of T
(X). So, R(X)
= R(X).



Proposition 5.6. Let R ⊆ S be an almost-Prüfer FCP extension, then
\ = R(X).
^
b
R(X)
= R(X)

\ = R(X)
^ = R(X)
e
b
Proof. We have a tower R(X) ⊆ R(X)
= R(X),
where the first and the third equalities come from Theorem 4.4 and
the second from Proposition 5.5.

We end this section with a special result.
Proposition 5.7. Let R ⊆ S be an extension such that R(X) ⊆ S(X)
\
b
has FIP, then R(X)
= R(X).

Proof. The map [R, S] → [R(X), S(X)] defined by T 7→ T (X) =
\ = T (X) for
R(X) ⊗R T is bijective [12, Theorem 32], whence R(X)
\ is a flat epimorphism.
b
some T ∈ [R, S]. Moreover, R(X)
→ R(X)
b = T and the result follows.
Since R → R(X) is faithfully flat, R

6. Fibers of quasi-Prüfer extensions
We intend to complete some results of Ayache-Dobbs [5]. We begin
by recalling some features about quasi-finite ring morphisms. A ring
morphism R → S is called quasi-finite by [40] if it is of finite type
and κ(P ) → κ(P ) ⊗R S is finite (as a κ(P )-vector space), for each
P ∈ Spec(R) [40, Proposition 3, p.40].

Proposition 6.1. A ring morphism of finite type is incomparable if
and only if it is quasi-finite and, if and only if its fibers are finite.
Proof. Use [41, Corollary 1.8] and the above definition.



30

G. PICAVET AND M. PICAVET

Theorem 6.2. An extension R ⊆ S is quasi-Prüfer if and only if
R ⊆ T is quasi-finite (respectively, has finite fibers) for each T ∈ [R, S]
such that T is of finite type over R, if and only if R ⊆ T has integral
fiber morphisms for each T ∈ [R, S].
Proof. It is clear that R ⊆ S is an INC-pair implies the condition
because of Proposition 6.1. To prove the converse, let T ∈ [R, S] and
write T as the union of its finite type R-subalgebras Tα . Now let
Q ⊆ Q′ be prime ideals of T , lying over a prime ideal P of R and
set Qα := Q ∩ Tα and Q′α := Q′ ∩ Tα . If R ⊆ Tα is quasi-finite, then
Qα = Q′α , so that Q = Q′ and then R ⊆ T is incomparable. The last
statement is Proposition 3.8.

Corollary 6.3. An integrally closed extension is Prüfer if and only if
each of its subextensions R ⊆ T of finite type has finite fibers.
Proof. It is enough to observe that the fibers of a (flat) epimorphism
have a cardinal ≤ 1, because an epimorphism is spectrally injective. 
A ring extension R ⊆ S is called strongly affine if each of its subextensions R ⊆ T is of finite type. The above considerations show that
in this case R ⊆ S is quasi-Prüfer if and only if each of its subextensions R ⊆ T has finite fibers. For example, an FCP extension is
strongly affine and quasi-Prüfer. We also are interested in extensions
R ⊆ S that are not necessarily strongly affine and such that each of its
subextensions R ⊆ T have finite fibers.
Next lemma will be useful, its proof is obvious.
Lemma 6.4. Let R ⊆ S be an extension and T ∈ [R, S]
(1) If T ⊆ S is spectrally injective and R ⊆ T has finite fibers, then
R ⊆ S has finite fibers.
(2) If R ⊆ T is spectrally injective, then T ⊆ S has finite fibers if
and only if R ⊆ S has finite fibers.
Remark 6.5. Let R ⊆ S be an almost-Prüfer extension, such that the
e ⊆ S has finite fibers and let P ∈ Spec(R).
integral extension T := R
The study of the finiteness of FibR,S (P ) can be reduced as follows. As
R ⊆ S is an epimorphism, because it is Prüfer, it is spectrally injective
(see Scholium A). The hypotheses of Proposition 4.5 hold. We examine
three cases. In case (R : R) 6⊆ P , it is well known that RP = (R)P so
that |FibR,S (P )| = 1, because R → S is spectrally injective. Suppose
now that (R : R) = P . From (R : R) = (T : S)∩R, we deduce that P is
lain over by some Q ∈ Spec(T ) and then FibR,R (P ) ∼
= FibT,S (Q). The
conclusion follows as above. Thus the remaining case is (R : R) ⊂ P

QUASI-PRÜFER EXTENSIONS

31

and we can assume that P T = T for if not FibR,R (P ) ∼
= FibT,S (Q) for
some Q ∈ Spec(T ) by Scholium A (1).
e⊆S
Proposition 6.6. Let R ⊆ S be an almost-Pr̈ufer extension. If R
eP : SP ) is a maximal ideal of R
eP for
has finite fiber morphisms and (R
e then R ⊆ R and R ⊆ S have finite fibers.
each P ∈ SuppR (S/R),
Proof. The Prüfer closure commutes with the localization at prime
e Let P be a prime ideal
ideals by Proposition 4.12. We set T := R.
of R and ϕ : R → RP the canonical morphism. We clearly have
FibR,. (P ) = a ϕ(FibRP ,.P (P RP )). Therefore, we can localize the data at
P and we can assume that R is local.
In case (T : S) = T , we get a factorization R → R → T . Since
R → T is Prüfer so is R → R and it follows that R = R because a
Prüfer extension is integrally closed.
From Proposition 1.2 applied to R ⊆ T , we get that there is some
P ∈ Spec(R) such that T = RP , R/P is a valuation ring with quotient
field T /P and P = PT . It follows that (T : S) = PT = P ⊆ R, and
hence (T : S) = (T : S) ∩ R = (R : R). We have therefore a pushout
diagram by Theorem 4.6:
R′ := R/P −−−→ R/P := R′




y
y
T ′ := T /P −−−→ S/P := S ′

where R/P is a valuation domain, T /P is its quotient field and R/P →
S/P is Prüfer by [26, Proposition 5.8, p. 52].
Because R′ → S ′ is injective and a flat epimorphism, there is a bijective map Min(S ′ ) → Min(R′ ). But T ′ → S ′ is the fiber at P of T → S
and is therefore finite. Therefore, Min(S ′ ) is a finite set {N1 , . . . , Nn }
of maximal ideals lying over the minimal prime ideals {M1 , . . . , Mn } of
R′ lying over 0 in R′ . We infer from Lemma 3.7 that R′ /Mi → S ′ /Ni
is Prüfer, whence integrally closed. Therefore, R′ /Mi is an integral domain and the integral closure of R′ in S ′ /Ni . Any maximal ideal M of
R′ contains some Mi . To conclude it is enough to use a result of Gilmer
[19, Corollary 20.3] because the number of maximal ideals in R′ /Mi is
less than the separable degree of the extension of fields T ′ ⊆ S ′ /Ni . 
e : S) is a maximal ideal of R.
e We
Remark 6.7. (1) Suppose that (R
e : S)P ⊆ (R
eP : SP ) and the hypotheses on (R
e : S) of
clearly have (R
the above proposition hold.

32

G. PICAVET AND M. PICAVET

e ⊆ S is a tower of finitely many integral minimal
(2) In case R
e =
extensions Ri−1 ⊆ Ri with Mi = (Ri−1 : Ri ), then SuppRe (S/R)
e where Ni = Mi ∩ R. If the ideals Ni are dif{N1 , . . . , Nn } ⊆ Max(R)
e ⊆ S is integral minimal and the
ferent, each localization at Ni of R
above result may apply. This generalizes the Ayache-Dobbs result [5,
e ⊆ S is supposed to be integral minimal.
Lemma 3.6], where R

Proposition 6.8. Let R ⊆ S be a quasi-Prüfer ring extension.
(1) R ⊆ S has finite fibers if and only if R ⊆ R has finite fibers.
(2) R ⊆ R has finite fibers if and only if each extension R ⊆ T ,
where T ∈ [R, S] has finite fibers.

Proof. (1) Let P ∈ Spec(R) and the morphisms κ(P ) → κ(P ) ⊗R R →
κ(P )⊗R S. The first (second) morphism is integral (a flat epimorphism)
because deduced by base change from the integral morphism R → R
(the flat epimorphism R → S). Therefore, the ring κ(P ) ⊗R R is zero
dimensional, so that the second morphism is surjective by Scholium
A (2). Set A := κ(P ) ⊗R R and B := κ(P ) ⊗R S, we thus have a
module finite flat ring morphism A → B. Hence, AQ → BQ is free for
each Q ∈ Spec(A) [16, Proposition 9] and BQ 6= 0 because it contains
κ(P ) 6= 0. Therefore, AQ → BQ is injective and it follows that A ∼
= B.
(2) Suppose that R ⊆ R has finite fibers and let T ∈ [R, S], then
R ⊆ RT is a flat epimorphism by Proposition 4.5(1) and so is κ(P ) ⊗R
R → κ(P ) ⊗R RT . Since Spec(κ(P ) ⊗R RT ) → Spec(κ(P ) ⊗R R)
is injective, R ⊆ RT has finite fibers. Now R ⊆ T has finite fibers
because T ⊆ RT is integral and is therefore spectrally surjective. 
Remark 6.9. Actually, the statement (1) is valid if we only suppose
that R ⊆ S is a flat epimorphism.
Next result contains [5, Lemma 3.6], gotten after a long proof.
Corollary 6.10. Let R ⊆ S be an almost-Prüfer extension. Then
R ⊆ S has finite fibers if and only if R ⊆ R has finite fibers, and if
e ⊆ S has finite fibers.
and only if R
Proof. By Proposition 6.8(1) the first equivalence is clear. The second
is a consequence of Lemma 6.4(2).

The following result is then clear.
Theorem 6.11. Let R ⊆ S be a quasi-Prüfer extension with finite
fibers, then R ⊆ T has finite fibers for each T ∈ [R, S].
Corollary 6.12. If R ⊆ S is quasi-finite and quasi-Prüfer, then R ⊆ T
e ⊆ S is module finite.
has finite fibers for each T ∈ [R, S] and R

QUASI-PRÜFER EXTENSIONS

33

Proof. By the Zariski Main Theorem, there is a factorization R ⊆ F ⊆
S where R ⊆ F is module finite and F ⊆ S is a flat epimorphism
[40, Corollaire 2, p.42]. To conclude, we use Scholium A in the rest of
e ⊗R F → S is injective because F → R
e ⊗R F
the proof. The map R
is a flat epimorphism and is surjective, since it is integral and a flat
e ⊗R F → S is a flat epimorphism .
epimorphism because R


Corollary 6.13. An FMC extension R ⊆ S is such that R ⊆ T has
finite fibers for each T ∈ [R, S].

Proof. Such an extension is quasi-finite and quasi-Prüfer. Then use
Corollary 6.12.

[5, Example 4.7] exhibits some FMC extension R ⊆ S, such that
R ⊆ R has not FCP. Actually, [R, R] is an infinite (maximal) chain.
Proposition 6.14. Let R ⊆ S be a quasi-Prüfer extension such that
R ⊆ R has finite fibers and R is semi-local. Then T is semi-local for
each T ∈ [R, S].
Proof. Obviously R is semi-local. From the tower R ⊆ T R ⊆ S we
deduce that R ⊆ T R is Prüfer. It follows that T R is semi-local [5,
Lemma 2.5 (f)]. As T ⊆ T R is integral, we get that T is semi-local. 
The following proposition gives a kind of converse.
Proposition 6.15. Let R ⊆ S be an extension with R semi-local. Then
R ⊆ S is quasi-Prüfer if and only if T is semi-local for each T ∈ [R, S].
Proof. If R ⊆ S is quasi-Prüfer, R ⊆ S is Prüfer. Let T ∈ [R, S]
and set T ′ := T R, so that T ⊆ T ′ is integral, and R ⊆ T ′ is Prüfer
(and then a normal pair). It follows from [5, Lemma 2.5 (f)] that T ′ is
semi-local, and so is T .
If T is semi-local for each T ∈ [R, S], so is any T ∈ [R, S]. Then,
(R, S) is a residually algebraic pair [6, Theorem 3.10] (generalized to
arbitrary extensions) and so is RM ⊆ SM for each M ∈ Max(R),
whence is Prüfer [6, Theorem 2.5] (same remark) and Proposition 1.2.
Then, R ⊆ S is Prüfer by Proposition 1.1 and R ⊆ S is quasi-Prüfer.

7. Numerical properties of FCP extensions
Lemma 7.1. Let R ⊂ S be an FCP extension. The map ϕ : [R, S] →
{(T ′, T ′′ ) ∈ [R, R]×[R, S] | SuppT ′ (R/T ′)∩SuppT ′ (T ′′ /R) = ∅}, defined
by ϕ(T ) := (T ∩ R, RT ) for each T ∈ [R, S], is bijective. In particular,
if R ⊂ S has FIP, then |[R, S]| ≤ |[R, R]||[R, S]|.

34

G. PICAVET AND M. PICAVET

Proof. Let (T ′ , T ′′) ∈ [R, R]×[R, S]. Then, R is also the integral closure
of T ′ in T ′′ (and in S).
Let T ∈ [R, S]. Set T ′ := T ∩ R and T ′′ := RT . Then (T ′ , T ′′ ) ∈
[R, R] × [R, S]. Assume that T ′ = T ′′ , so that T ′ = T ′′ = R, giving T =
R and SuppT ′ (R/T ′ ) = SuppT ′ (T ′′ /R) = ∅. Assume that T ′ 6= T ′′ . In
view of [39, Proposition 3.6], we get SuppT ′ (R/T ′ )∩SuppT ′ (T ′′ /R) = ∅.
Hence ϕ is well defined.
Now, let T1 , T2 ∈ [R, S] be such that ϕ(T1 ) = ϕ(T2 ) = (T ′ , T ′′).
Assume T ′ 6= T ′′ . Another use of [39, Proposition 3.6] gives that T1 =
T2 . If T ′ = T ′′ , then, T ′ = T ′′ = R, so that T1 = T2 = R. It follows
that ϕ is injective. The same reference gives that ϕ is bijective.

Proposition 7.2. Let R ⊂ S be a FCP extension. We define two
order-isomorphisms ϕ′ and ψ as follows:
~ → [R, R] × [R, R]
~ defined by ϕ′ (T ) := (T ∩ R, T R)
ϕ′ : [R, R]
~ → [R, R]
e × [R,
e R]
~ defined by ψ(T ) := (T ∩ R,
e T R).
e
ψ : [R, R]

Proof. This follows from [39, Lemma 3.7] and Proposition 4.19. (We
~ = R.)
e
recall that R


~ R)
e = Supp(R/R),
Corollary 7.3. If R ⊆ S has FCP, then Supp(R/
e
~
e
~
= Supp(R/R)
and Supp(R/R)
= Supp(R/R)∪Supp(R/R).
Supp(R/R)

e R),
e B := Supp(R/R),
e
e
Proof. Set A := Supp(R/
C := Supp(R/R)
e
with
and D := Supp(R/R). Then, A ∪ B = C ∪ D = Supp(R/R),
A ∩ B = C ∩ D = B ∩ D = ∅ by Proposition 4.19.
Assume that A ∪ B 6= B ∪ D and let P ∈ (A ∪ B) \ (B ∪ D).
e P = (R)P (R)
e P = RP , a contradiction. It follows that
Then, RP 6= (R)
A ∪ B = B ∪ D. Intersecting the two members of this equality with
A and D, we get A = A ∩ D = D. In the same way, intersecting the
equality A ∪ B = C ∪ D = C ∪ A by B and C, we get B = C.

Corollary 7.4. Let R ⊂ S be an FCP extension. We define two orderisomorphisms
~ by ϕ1 (T ) := T R
e → [R, R]
ϕ1 : [R, R]
e R]
~ by ψ1 (T ) := T R.
e
ψ1 : [R, R] → [R,
Proof. We use notation of Proposition 4.19. We begin to remark that
e play symmetric roles.
R and R
e be such that ϕ1 (T ) = ϕ1 (T ′ ). Since T ∩ R =
Let T, T ′ ∈ [R, R]
′
T ∩ R = R by Proposition 4.19, we get ϕ(T ) = ϕ(T ′ ), so that T = T ′
and ϕ1 is injective. A similar argument shows that ψ1 is injective.

QUASI-PRÜFER EXTENSIONS

35

e There exists T ∈ [R, R]
e such that ϕ(T ) = (R, U), so
Let U ∈ [R, R].
e
e
that R = T ∩ R and U = T R. Let M ∈ Supp(R/R)
= Supp(R/R)
∪
e
Supp(R/R) by Corollary 7.3. If M ∈ Supp(R/R),
then M 6∈ Supp(R/R)

e M = RM R
eM = R
eM . If M ∈
by Proposition 4.19, giving TM ⊆ R
Supp(R/R), the same reasoning gives TM ⊆ RM , so that RM = TM ∩
eM . Then, TM = R
eM . It follows that T ⊆ R,
e
RM = TM , but RM = R
e
giving T ∈ [R, R] and ϕ1 is surjective, hence bijective. A similar argument shows that ψ1 is surjective, hence bijective.


~
e × [R, R] → [R, R]
Corollary 7.5. If R ⊂ S has FCP, then θ : [R, R]
′
′
defined by θ(T, T ) := T T , is an order-isomorphism. In particular, if
e
~ ≤ |[R, S]|.
R ⊂ S has FIP, then |[R, R]||[R,
R] = |[R, R]k
Proof. Using notation of Proposition 7.2 and Corollary 7.4, we may
remark that ψ◦θ = Id×ψ1 . Since ψ and Id×ψ1 are order-isomorphisms,
so is θ. The FIP case is obvious.

Gathering the previous results, we get the following theorem.
Theorem 7.6. If R ⊂ S has FCP, the next statements are equivalent:
(1) Supp(R/R) ∩ Supp(S/R) = ∅.
(2) The map ϕ : [R, S] → [R, R] × [R, S] defined by ϕ(T ) := (T ∩
R, T R) is an order-isomorphism.
(3) R ⊆ S is almost-Prüfer.
e
(4) Supp(S/R) = Supp(R/R).
e → [R, S] defined by ϕ1 (T ) := T R is an
(5) The map ϕ1 : [R, R]
order-isomorphism.
e S] defined by ψ1 (T ) := T R
e is an
(6) The map ψ1 : [R, R] → [R,
order-isomorphism.
e × [R, R] → [R, S] defined by θ(T, T ′ ) := T T ′
(7) The map θ : [R, R]
is an order-isomorphism.
e = Supp(R/R).
If one of these conditions holds, then Supp(S/R)
If R ⊂ S has FIP, the former conditions are equivalent to each of
the following conditions:
e
R]|.
(8) |[R, S]| = |[R, R]||[R,
(9) k[R, S]| = |[R, R]||[R, S]|.
e = |[R, S]|.
(10) |[R, R]|
e S]|.
(11) |[R, R]| = |[R,
Proof. (1) ⇒ (2) by [39, Lemma 3.7].

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G. PICAVET AND M. PICAVET

(2) ⇒ (1). If the statement (2) holds, there exists T ∈ [R, S] such
that T ∩ R = R and T R = S. Then, [39, Proposition 3.6] gives that
Supp(R/R) ∩ Supp(S/R) = ∅.
(1) ⇒ (3) by [39, Proposition 3.6].
(3) ⇒ (4), (5), (6) and (7): Use Corollary 7.3 to get (4), Corollary 7.4
to get (5) and (6), and Corollary 7.5 to get (7). Moreover, (3) and
e = Supp(R/R).
Corollary 7.3 give Supp(S/R)
(4) ⇒ (1) by Proposition 4.19(2).
e
(5), (6) or (7) ⇒ (3) because, in each case, we have S = RR.
Assume now that R ⊂ S has FIP.
Then, obviously, (7) ⇒ (8), (2) ⇒ (9), (5) ⇒ (10) and (6) ⇒ (11).
e so
e
R]| = |[R, R]|,
(9) ⇒ (3) by Corollary 7.5, which gives |[R, R]||[R,

e and then S = R.
e
that |[R, S]| = |[R, R]|,
(8) ⇒ (1): Using the map ϕ of Lemma 7.1, we get that {(T ′ , T ′′) ∈
[R, R] × [R, S] | SuppT ′ (R/T ′ ) ∩ SuppT ′ (T ′′ /R) = ∅} = [R, R] × [R, S],
so that SuppR (R/R) ∩ SuppR (S/R) = ∅.
(10) ⇒ (3) and (11) ⇒ (3) by Corollary 7.4.


Example 7.7. We give an example where the results of Theorem 7.6
do not hold if R ⊆ S has not FCP. Set R := ZP and S := Q[X]/(X 2 ),
e = Q because R ⊂ R
e is Prüfer (minimal)
where P ∈ Max(Z). Then, R
e ⊂ S is integral minimal. Set M := P RP ∈ Max(R) with (R, M)
and R
a local ring. It follows that M ∈ Supp(R/R) ∩ Supp(S/R) because
R ⊂ S is neither integral, nor Prüfer. Similarly, M ∈ Supp(R/R) ∩
e
Supp(R/R).
Indeed, R ⊂ R has not FCP.
We end the paper by some length computations in the FCP case.

Proposition 7.8. Let R ⊆ S be an FCP extension. The following
statements hold:
e = ℓ[R, R]
~ and ℓ[R, R] = ℓ[R,
e R]
~
(1) ℓ[R, R]
~ = ℓ[R, R]
e + ℓ[R,
e R]
~ = ℓ[R, R] + ℓ[R, R]
~
(2) ℓ[R, R]
~ = |Supp (R/R)|
e = |SuppR (R/R)|.
e
~
(3) ℓ[R, R]
= ℓ[R, R]
R
Proof. To prove (1), use the maps ϕ1 and ψ1 of Corollary 7.4. Then (2)
follows from [11, Theorem 4.11] and (3) from [10, Proposition 6.12]. 
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Université Blaise Pascal, Laboratoire de Mathématiques, UMR6620
CNRS, 24, avenue des Landais, BP 80026, 63177 Aubière CEDEX, France
E-mail address: Gabriel.Picavet@math.univ-bpclermont.fr
E-mail address: picavet.gm(at)wanadoo.fr

