arXiv:1501.03411v4 [] 10 Dec 2015

PERINORMALITY – A GENERALIZATION OF KRULL
DOMAINS
NEIL EPSTEIN AND JAY SHAPIRO
Abstract. We introduce a new class of integral domains, the perinormal domains, which fall strictly between Krull domains and weakly normal domains. We establish basic properties of the class, and in the case
of universally catenary domains we give equivalent characterizations of
perinormality. (Later on, we point out some subtleties that occur only in
the non-Noetherian context.) We also introduce and explore briefly the
related concept of global perinormality, including a relationship with divisor class groups. Throughout, we provide illuminating examples from
algebra, geometry, and number theory.

1. Introduction
Motivated in part by the classical concept of a ring extension satisfying
going-down from Cohen and Seidenberg [CS46], the concept of the goingdown domain has been fruitful in non-Noetherian commutative ring theory
(see for example [Dob73, Dob74, DP76]); for Noetherian rings it merely
coincides with domains of dimension ≤ 1 [Dob73, Proposition 7]. By definition, a ring extension R ⊆ S satisfies going-down if whenever p ⊂ q are
prime ideals of R and Q ∈ Spec S with Q ∩ R = q, there is some prime
ideal P ∈ Spec S with P ⊂ Q and P ∩ R = p (a condition that is satisfied
whenever S is flat over R). Then an integral domain R is a going-down
domain if for every (local) overring S of R, the inclusion R ⊆ S satisfies
going-down. (In fact by [DP76, Theorem 1] it doesn’t matter whether one
specifies ‘local’ or not.)
It is natural to ask which overrings of an integral domain R satisfy goingdown over it. It is classical that any flat R-algebra (hence any flat overring)
will satisfy going-down over R [Mat86, Theorem 9.5]. Moreover, the flat
local overrings are precisely the rings Rp where p is a prime ideal of R
[Ric65, Theorem 2]. In this context, since going-down domains have proven
to be a useful concept, it makes sense to explore the orthogonal concept:
• When does it happen that the only local overrings that satisfy goingdown over R are the localizations at prime ideals?
Date: December 11, 2015.
2010 Mathematics Subject Classification. 13B21, 13F05, 13F45.
Key words and phrases. Krull domain, going down, perinormal, globally perinormal,
universally catenary.
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NEIL EPSTEIN AND JAY SHAPIRO

We call such a ring perinormal, and it is the subject of this paper. (The
related concept of global perinormality stipulates that the only overrings,
local or not, that satisfy going-down over the base are localizations of the
base ring at multiplicative sets.)
It turns out that the class of perinormal rings is closely related to Krull
domains (and so Noetherian normal domains) and weakly normal (hence
seminormal) domains in that Krull domain ⇒ perinormal ⇒ weakly normal and (R1 ), with neither implication reversible. Moreover, for universally
catenary domains (and somewhat more generally), we can characterize perinormal domains as those domains R such that no prime localization Rp has
an overring that induces a bijection on prime spectra. When R is smooth
in codimension 1, we can restrict our attention to integral overrings of these
Rp (cf. Theorem 4.7). On the other hand, the only perinormal going-down
domains are Prüfer domains.
The structure of the paper is as follows. We start by establishing some
basic facts in Section 2, including Proposition 2.5 which shows that perinormality is a local property and Proposition 2.4, which reframes perinormality
in terms of flatness. Section 3 explores the relationship of perinormality to
(generalized) Krull domains, weakly normal domains, and (R1 ) domains.
Theorem 3.10, Corollary 3.4, and Proposition 3.2 respectively show that
perinormality is implied by the first and implies the latter two properties. We also exhibit some sharpening examples. Section 4 is dedicated to
Theorem 4.7, which gives the two characterizations of perinormal domains
among the Noetherian domains mentioned above. In Section 5, we find Theorem 5.2, which exhibits a method for producing perinormal domains that
are not integrally closed. Section 6 is devoted to the related notion of global
perinormality; in particular, we give a partial characterization (see Theorem 6.4) of which Krull domains may be globally perinormal, along with
examples relevant to algebraic number theory. It turns out that the theory
of perinormality is a bit different when one includes non-Noetherian rings;
in Section 7, we point out the subtleties in a series of examples, including
the fact that not every integrally closed domain is perinormal (unlike in the
Noetherian case). We end with a list of interesting questions in Section 8.
Conventions: All rings are commutative with identity, and ring homomorphisms and containments preserve the multiplicative identity. The term
local means only that the ring has a unique maximal ideal. An overring of
an integral domain R is a ring sitting between R and its fraction field.

2. First properties
Definition 2.1. Let R be an integral domain. We say R is perinormal if
whenever S is a local overring of R such that the inclusion R ⊆ S satisfies
going-down, it follows that S is a localization of R (necessarily at a prime
ideal).

PERINORMALITY – A GENERALIZATION OF KRULL DOMAINS

3

We say that R is globally perinormal if the same conclusion holds when
the condition on S being local is dropped (so that this time, the localization
is just at a multiplicative set).
Remark. The term perinormal is meant to reflect several aspects of the
property: (1) It is closely related to the properties of seminormality and
weak normality (cf. Corollary 3.4). (2) It is closely related to the concept of
normality in the Noetherian case (cf. Theorem 3.10), which is usually the
only situation where the word “normal” is used for integral closedness in the
literature. (3) Perinormality is not a weakening of the property of integral
closedness in general (cf. Example 7.1), whence the prefix peri- (unlike weak
and semi-, which both imply weakenings).
Lemma 2.2. A homomorphism of commutative rings R → S satisfies goingdown if and only if for all P ∈ Spec S, the induced map Spec (SP ) →
Spec (RP ∩R ) is surjective.
Proof. This follows immediately from the definition.



Recall the following theorem of Richman.
Theorem 2.3 ([Ric65, Theorem 2]). Let A be an integral domain and B
an overring of A. Then B is flat over A if and only if Bm = Am∩A for all
maximal ideals m of B.
This theorem allows us to characterize perinormality in terms of flatness:
Proposition 2.4. A domain R is perinormal if and only if every overring
of R that satisfies going-down is flat over R.
Proof. Suppose R is perinormal. Let S be an overring of R that satisfies
going-down over R. Let m be a maximal ideal of R. Clearly Sm satisfies
going-down over R (since going-downness is transitive and Sm is going-down
over S), so by perinormality, Sm is a localization of R, so that necessarily,
Sm = Rm∩R . Then by Theorem 2.3, S is flat over R.
Conversely, suppose every overring of R that satisfies going-down is flat
over R. Let (S, m) be a local overring of R that satisfies going-down. Then
it is flat over R, so again by Theorem 2.3, S = Rm∩R is a localization of R.
Thus, R is perinormal.

Next, we show that perinormality is a local property.
Proposition 2.5. If R is perinormal, so is RW for every multiplicative set
W . Conversely, if Rm is perinormal for all maximal ideals m of R, then so
is R.
Proof. First suppose R is perinormal. Let S be a local overring of RW that
satisfies going-down. Then S satisfies going-down over R (since no prime
ideal of R lain over by a prime of S can intersect W ) and so S = RV for some
multiplicatively closed subset V of R. But then V is also a multiplicatively
closed subset of RW , and S = (RW )V . Therefore RW is perinormal.

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NEIL EPSTEIN AND JAY SHAPIRO

Conversely, suppose that Rm is perinormal for all maximal ideals m of R.
Let (S, n) be a local overring of R such that the inclusion R ⊆ S satisfies
going-down. Let m be a maximal ideal of R such that n∩R ⊆ m. Then Rm ⊆
S satisfies going-down, so that by perinormality of Rm , S = (Rm )n∩Rm =
Rn∩R . Thus R is perinormal.

Example 2.6. Any valuation domain R is globally perinormal because every
overring of R is a localization, as is easily shown. It then follows from
Proposition 2.5 that every Prüfer domain is perinormal.
3. (R1 ) domains, weakly normal domains, and generalized
Krull domains
In this section, we fit perinormality into the context of three known important classes of integral domains. Namely, generalized Krull =⇒ Krull =⇒
perinormal =⇒ weakly normal and (R1 ), with neither arrow reversible.
Definition 3.1. We say that a commutative ring R satisfies (R1 ) if RP is
a valuation domain whenever P is a height one prime of R.
Remark. It seems that in the literature, the term (R1 ) is only used for
Noetherian rings (cf. [Mat86, p. 183]). Here we have extended it to arbitrary commutative rings in a way that both coincides with the established
definition in the Noetherian case and suits our purpose in the general case.
Proposition 3.2. Any perinormal domain R satisfies (R1 ).
Proof. Let p be a height one prime of R. Let (V, m) be a valuation overring
of R such that m ∩ R = p. (If R is Noetherian, we can choose V to be
Noetherian as well.) Then the map R → V trivially satisfies going-down.
Thus, V is a localization of R, whence V = Rm∩R = Rp , completing the
proof that R satisfies (R1 ).

Proposition 3.3. If (R, m) is a local perinormal domain, then for any integral overring S of R such that Spec S → Spec R is a bijection, we have
R = S.
Proof. Let S be an integral overring of R such that Spec S → Spec R is
bijective. By integrality of the extension, some prime ideal n of S lies over
m; by bijectivity, there can be only one such prime; since fibers of Spec maps
on integral extensions are antichains, n is maximal, and the unique maximal
ideal of S.
Now, let p ⊂ q be a chain of primes in R, and let Q ∈ Spec S with
Q ∩ R = q. Since the Spec map is surjective, there is some P ∈ Spec S with
P ∩ R = p. By the ‘going up’ property of integral extensions, there is some
Q′ ∈ Spec S such that P ⊆ Q′ and Q′ ∩ R = q. But then by injectivity
of the Spec map, Q′ = Q. This shows that the inclusion R ⊆ S satisfies
going-down; hence S is a localization of R since R is perinormal. But since
the map R → S is a local homomorphism of local rings, the only way S can
be a localization of R is if R = S.


PERINORMALITY – A GENERALIZATION OF KRULL DOMAINS

5

Recall that an integral domain R is weakly normal 1 if for any integral
overring S of R such that the map Spec S → Spec R is a bijection and for
all P ∈ Spec S (where we set p := P ∩ R), the corresponding field extensions
Rp /pRp → SP /P SP is purely inseparable, it follows that R = S.
A domain R is seminormal if [Swa80] whenever x is an element of the
fraction field with x2 , x3 ∈ R, we have x ∈ R. However, it is equivalent
to say that for any integral overring S such that Spec S → Spec R is a
bijection and the corresponding field extensions Rp /pRp → SP /P SP are
isomorphisms, then R = S. From this, it is clear that every weakly normal
domain is seminormal, and that for a domain that contains Q, the converse
holds.
Recall that both weak normality and seminormality are local properties
in the sense of Proposition 2.5. Also every normal domain is weakly normal.
For all this and more, cf. Vitulli’s survey article on weak normality and
seminormality [Vit11].
Corollary 3.4. If R is perinormal, then it is weakly normal (hence seminormal).
Proof. Since both perinormality and weak normality are local properties,
we may assume R is local. Now let S be an integral overring of R where
Spec S → Spec R is a bijection such that for any P ∈ Spec S, the field
extension Rp /pRp → SP /P SP is purely inseparable (where p = P ∩ R).
Then by Proposition 3.3, R = S. It follows that R is weakly normal.

We next present two examples to show that the converse to Corollary 3.4
is false, even under some additional restrictions.
Example 3.5. Not all weakly normal (resp. seminormal) domains are perinormal, even in dimension 1. For example, A = R[x, ix] is seminormal, even
weakly normal, without being perinormal. Failure of perinormality arises
from the fact that C[x] is going-down over A (with the same fraction field
C(x)) without being a localization of it. To see seminormality, merely observe that A consists of those polynomials whose constant term is real, and
if f ∈ C[x] is such that its square and cube have real constant term, it follows that the constant term of f has its square and cube in R, whence the
constant term of f is in R already.
Example 3.6. (Thanks to Karl Schwede for this example.) Even for finitely
generated algebras over algebraically closed fields, weakly normal (R1 ) domains are not necessarily perinormal. For an example, consider R = k[x, y, xz, yz, z 2 ]
where k is any field of characteristic not equal to 2. Let A = k[x, y, z], and
note that A is the integral closure of R. Hence every prime ideal of R is
contracted from A. Let P ∈ Spec A. If P + (x, y), then z ∈ RP ∩R , whence
RP ∩R = AP is regular. Therefore, RP ∩R is normal, weakly normal, and
perinormal. This also shows that R satisfies (R1 ).
1This is not the original definition [AB69], but it is equivalent [Yan83, Remark 1].

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NEIL EPSTEIN AND JAY SHAPIRO

Further, Yanagihara [Yan83, Proposition 1] has shown that an arbitrary
pullback of a weakly normal inclusion is also a weakly normal inclusion.
Hence, we may conclude that R is weakly normal, as R/I is a subring of
A/I, where I = (x, y, xz, yz)R = (x, y)A, and the extension k[z 2 ] ֒→ k[z]
is weakly normal, since char k 6= 2. (One may similarly show the ring is
seminormal even when char k = 2 by using [GT80, 4.3] in place of [Yan83,
Proposition 1], which shows the analogous fact for seminormal inclusions.)
However, the ring R(x,y)∩R = k(z 2 )[x, y, xz, yz](x,y,xz,yz) is not perinormal.
Its integral closure is A(x,y) = k(z)[x, y](x,y) . Then the map R(x,y)∩R →
A(x,y) induces a bijection on spectra because for all other primes, we have
an isomorphism, whereas the localness of the integral closure shows that we
also have bijectivity at the maximal ideal. But the two rings are unequal
because z ∈
/ R(x,y)∩R . Then since R(x,y)∩R is not perinormal, neither is R.
Lemma 3.7. Let R be an integral domain, S an overring of R, and p ∈
Spec S such that V := Rp∩R is a valuation domain of dimension 1. Then
Rp∩R = Sp as subrings of the fraction field K of R, and ht p = 1.
Proof. We have V = Rp∩R ⊆ Sp ⊆ K. But Sp 6= K, since p 6= 0. On the
other hand, V is a valuation domain, so every overring is a localization at a
prime ideal. Since V has only two primes, the only possibilities are V and
K. Since Sp 6= K, it follows that Sp = V . Finally, ht p = dim Sp = dim V =
1.

Definition 3.8. For a commutative ring R, Spec 1 (R) denotes the set of all
height one primes of R.
Proposition 3.9. Let R be an (R1 ) domain and let S be an overring such
that the extension R ⊆ S satisfies going-down. Then S satisfies (R1 ), and
the map Spec S → Spec R induces an injective map Spec 1 (S) → Spec 1 (R)
whose image consists of those height one primes p of R such that pS 6= S.
Proof. First we need to show that given a height one prime Q of S, q := Q∩R
is a height one prime of R. We have q 6= 0 because R ⊆ S is an essential
extension of R-modules; hence ht q ≥ 1. On the other hand, suppose there
is some p ∈ Spec R with 0 ( p ( q. Then by going-down, there is some
P ∈ Spec S with P ∩ R = p. But then P 6= 0 (again by essentiality of the
extension), whence 0 ( P ( Q is a chain of primes in S, so that ht Q ≥ 2,
a contradiction. Then by Lemma 3.7, S satisfies (R1 ).
Next, let p ∈ Spec 1 (R). If pS = S, then no prime of S can lie over p.
On the other hand, if pS 6= S, then there is some maximal ideal Q of S
with pS ⊆ Q. Then the going-down property implies that there is some
P ∈ Spec S with P ∩ R = p. Moreover, Lemma 3.7 along with the (R1 )-ness
of R implies that SP = Rp and ht P = 1. Finally, if there is some other
prime ideal P ′ of S with P ′ ∩ S = p, then we have SP = Rp = SP ′ . But
different prime ideals of a ring always give rise to different localizations, so
P = P ′ , finishing the proof that the map of Spec 1 ’s is injective.


PERINORMALITY – A GENERALIZATION OF KRULL DOMAINS

7

Consider the following properties that an integral domain R may have:
T
(1) R = p∈Spec 1 (R) Rp .
(2) For any nonzero element r ∈ R, the set {p ∈ Spec 1 (R) | r ∈ p} is
finite.
(3) Rp is a DVR for all p ∈ Spec 1 (R).
One says R is a Krull domain (resp. generalized Krull domain) if it satisfies
properties (1–3) (resp. properties (1), (2), and (R1 )).
Recall the Mori-Nagata theorem (cf. [Fos73, Theorem 4.3]), which says
that the integral closure of any Noetherian domain is Krull (though not
necessarily Noetherian); hence every Noetherian normal domain is Krull.
Theorem 3.10. If R is a generalized Krull domain (e.g. Noetherian normal),
then R is perinormal.
Proof. Let (S, m) be a local overring of R such that the inclusion R ⊆ S
satisfies going-down. Let Q = m ∩ R; RQ is then also a generalized Krull
domain [Gil72, Corollary 43.6]. Note that the going-down condition implies
that the map Spec S → Spec RQ is surjective. Hence by Proposition 3.9, we
get a bijective map Spec 1 (S) → Spec 1 (RQ ), and for each P ∈ Spec 1 (S) and
corresponding p = P ∩ R ∈ Spec 1 (RQ ), we have (RQ )p = SP by Lemma 3.7.
Therefore
\
\
RQ ⊆ S ⊆
SP =
(RQ )p = RQ .
P ∈Spec 1 (S)

p∈Spec 1 (RQ )

That is, S = RQ , so R is perinormal.



4. Local characterizations of perinormality
In this section, after a preliminary exploration of how (R1 ) domains interact with overrings and the special relationship that occurs between two
rings that share a nonzero ideal, we provide two surprising characterizations
of perinormal domains within a large class of integral domains.
Lemma 4.1. Let R be an (R1 ) integral domain whose integral closure R′
is a generalized Krull domain and such that for all P ∈ Spec 1 R′ , P ∩ R ∈
Spec 1 R. If there is a maximal ideal of R that contains all the height one
primes of R, then R is local.
Proof. Let m be a maximal ideal of R, and suppose that m contains all height
one primes of R. By Lemma 3.7, since R is an (R 1 ) domain, if p ∈ Spec 1 R
and P ∈ Spec 1 R′ lies over p, then Rp = RP′ . As height one primes of R′
contract to height one primes of R, we have
\
\
\
Rm ⊆
(Rm )p =
RP =
RP = R ′ ,
p∈Spec 1 (Rm )

P ⊆m,ht P =1

P ∈Spec 1 R

where the last equality follows since R′ is generalized Krull. We have shown
that Rm is integral over R, which can only happen if R = Rm .


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NEIL EPSTEIN AND JAY SHAPIRO

Lemma 4.2. Let (R, m) be an (R1 ) local domain whose integral closure R′
is a generalized Krull domain and such that for all P ∈ Spec 1 R′ , P ∩ R ∈
Spec 1 R. Let S be an integral overring of R that satisfies going-down over
R. Then S is local.
Proof. By Proposition 3.9, S satisfies (R1 ) and the map Spec S → Spec R
∼
induces a bijection Spec 1 (S) → Spec 1 (R). Now let n be a maximal ideal of S
that contains mS; it exists because S is integral over R. Then the extension
R ⊆ Sn is going-down and mSn 6= Sn , so Proposition 3.9 applies again to
∼
produce a bijection Spec 1 (Sn ) → Spec 1 (R). This bijection composes with
the inverse of the previous bijection to give a bijection of Spec 1 (Sn ) with
Spec 1 (S). Hence, for all height one primes p of S, we have pSn 6= Sn –
that is, p ⊆ n. Moreover, R and S have the same integral closure, which
is generalized Krull by hypothesis. As height one primes of R′ contract to
height one primes of R, one can show using the properties of integrality that
the same holds for all intermediate rings. Thus by Lemma 4.1, S must be
local.

Next we give conditions on a domain A that ensure that height one primes
of A′ contract to height one primes of A. We first need to recall some
definitions. A ring A is called catenary if given a pair p1 ⊂ p2 of prime ideals
of A such that there exists a saturated chain of prime ideals between the
two, then all such saturated chains have the same length. We say that A is
universally catenary if it is Noetherian and every finitely generated A-algebra
is catenary. It is clear that being catenary and hence being universally
catenary is closed under localization.
Let A ⊆ B be integral domains. Then tr.degA B denotes the transcendence degree of the fraction field of B over that of A. Recall that the ring
extension is said to satisfy the dimension (or altitude) formula if the following equality holds for all P ∈ Spec B:
htP + tr.degA/p B/P = ht p + tr.degA B
where p = P ∩ A (see for example [Mat86, p. 119]). We note that if in
addition B is integral over A, then tr.degA B = 0 = tr.degA/p B/P in which
case the height of a prime of B is invariant under contraction to A.
Lemma 4.3. Let R be a universally catenary integral domain with integral
closure R′ . Then every height one prime ideal of R′ contracts to a height
one prime ideal of R.
Proof. Since R is Noetherian, by [Rat80, Corollary 2.5] it will suffice to
show that if f ∈ R[x]′ , then the height of a prime ideal of R[x, f ] is invariant
under contraction to R[x]. Since R[x] is also universally catenary and R[x, f ]
is module finite over R[x] (in particular algbera finite), it follows that the
extension R[x] ⊆ R[x, f ] satisfies the dimension formula (see for example
[Mat86, Theorem 15.6]). Since it is also an integral extension, we have that
the height is invariant under contraction as desired.


PERINORMALITY – A GENERALIZATION OF KRULL DOMAINS

9

Remark. The universal catenarity assumption is not particularly restrictive, as almost every Noetherian ring that arises in algebraic geometry,
number theory, and everyday commutative algebra is universally catenary.
Indeed, the class of universally catenary rings is closed under localization
and finitely generated algebra extensions. Moreover, it includes CohenMacaulay rings (including Dedekind domains, fields, and regular local rings;
see [Mat86, Theorem 17.9]) and complete Noetherian local rings (e.g. power
series rings in finitely many variables over a field or over the p-adics; see
[Mat86, Theorem 29.4]).
The following three results are well-known to experts, and some of their
statements appear (without proofs) in [Fon80]. However, we include them
here for completeness and to make the paper self-contained.
Lemma 4.4. Let R ⊆ T be an inclusion of commutative rings, and let I
be an ideal that is common to R and T . (That is, I is an ideal of R and
IT = I.) Let W be a multiplicatively closed subset of T , set V := W ∩R, and
suppose that I ∩W 6= ∅. Then the natural map RV → TW is an isomorphism.
Proof. Let z ∈ I ∩ W . To see injectivity, let vr ∈ RV (with r ∈ R, v ∈ V )
such that vr = 0 in TW . Then for some w ∈ W , we have wr = 0. Moreover,
zw ∈ I ∩ W ⊆ R ∩ W = V and (zw)r = 0, whence vr = 0 in RV .
To see surjectivity, let wt ∈ TW (with t ∈ T , w ∈ W ). Then zt ∈ IT ⊆ R
zt
and zw ∈ I ∩ W ⊆ R ∩ W = V , so that wt = zw
∈ RV .

Corollary 4.5. Let R ⊆ T be an inclusion of commutative rings, and let I
be an ideal common to R and T . Let z ∈ I, and let P ∈ Spec T with I * P .
Then the natural maps Rz → Tz and RP ∩R → TP are isomorphisms.
Proof. In the first case, apply Lemma 4.4 with V = W = {z n | n ∈ N}. In
the second case, apply the same lemma with W = T \ P .

Corollary 4.6. Let R ⊆ T be integral domains that share a common nonzero
ideal I. Then the induced map of fraction fields is an isomorphism.
Proof. Apply Lemma 4.4 with W = T \ {0}.



Theorem 4.7. Let R be a universally catenary integral domain with fraction
field K. The following are equivalent.
(a) R is perinormal.
(b) For each p ∈ Spec R, Rp is the only ring S between Rp and K such
that the induced map Spec S → Spec Rp is an order-reflecting bijection.
(c) R satisfies (R1 ), and for each p ∈ Spec R, Rp is the only ring
S between Rp and its integral closure such that the induced map
Spec S → Spec Rp is an order-reflecting bijection.
Proof. We note that we only need the universal catenarity condition for the
implication (c) =⇒ (a).

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NEIL EPSTEIN AND JAY SHAPIRO

(a) =⇒ (b): Since perinormality localizes, we may assume that (R, p) is
local. Now let S be a ring between R and K such that Spec S → Spec R is
an order-reflecting bijection. Thus S satisfies going-down over R. Since S
is local, the perinormality assumption on R implies that S is a localization
of R. As the Spec map is onto, we conclude that R = S.
(b) =⇒ (c): To see that R satisfies (R1 ), let p be a height one prime
of R. Let V be a valuation ring centered on p. Then all nonzero prime
ideals of V contract to p, and their intersection q is also a prime ideal of V .
Since q contains no prime ideals other than itself and (0), we have ht q = 1.
Now, the map Rp → Vq induces a bijection on Spec, so Rp = Vq , a valuation
domain. On the other hand, the second condition in (c) follows trivially
from (b).
(c) =⇒ (a): Let (S, n) be a going-down local overring of R. Let p =
n ∩ R. Note that Rp satisfies (R1 ), so that by Proposition 3.9, the map
∼
Spec S → Spec Rp induces a bijection Spec 1 (S) → Spec 1 (Rp ) where by
Lemma 3.7, the corresponding localizations of S and Rp coincide. Hence S
is (R1 ). Since Rp is also universally catenary it follows by integrality and
Lemma 4.3 that the Spec map Spec (Rp )′ → Spec Rp induces a bijection
∼
Spec 1 (Rp )′ → Spec 1 Rp , where again the corresponding localizations of Rp
and (Rp )′ coincide. Thus,
\
\
Rp ⊆ S ⊆
SQ =
(Rp )P = (Rp )′ ,
Q∈Spec 1 (S)

P ∈Spec 1 (Rp )

where the last equality follows since (Rp )′ is a Krull domain. Hence S is
integral over Rp .
Next, we claim that the map Spec S → Spec Rp is injective. To see this,
let Q be a prime ideal of Rp , and let W := Rp \ Q. Then the inclusion
(Rp )Q ⊆ SW is integral, it satisfies going-down, and QSW 6= SW . Moreover,
′
the integral closure of Rp is RR\p
, a Krull domain. Thus by Lemmas 4.2
and 4.3, SW is local. But this means that only one prime of S lies over Q,
whence the map Spec S → Spec Rp is injective.
However, the map is also surjective, since S is integral over Rp . Therefore
the map is bijective.
Finally, we must show that the map is order-reflecting – that is, if q1 ⊆
q2 in Rp , then the corresponding primes in S are also so ordered. So let
Qj ∈ Spec S with Qj ∩ Rp = qj , j = 1, 2. By going-down, there is some
P ∈ Spec S with P ⊆ Q2 and P ∩ Rp = q1 = Q1 ∩ Rp . But then by the
injectivity of the Spec map, P = Q1 , whence Q1 ⊆ Q2 . Hence, condition (c)
applies and Rp = S, whence R is perinormal.

Recall [FO70] that a ring extension A ⊆ B is called minimal if there are
no rings properly between A and B.
Corollary 4.8. Let (R, m) be a universally catenary local domain. Assume
that dim R ≥ 2 and that the map R → S is a minimal ring extension, where

PERINORMALITY – A GENERALIZATION OF KRULL DOMAINS

11

S is the integral closure of R. Then R is perinormal if and only if S is not
local.
Proof. By [FO70, Theorem 2.2], m is also an ideal of S. Now let p ∈ Spec R
with p 6= m. Let P, P ′ ∈ Spec S with P ∩ R = P ′ ∩ R = p. Then by
Corollary 4.5, SP = Rp = SP ′ , whence P = P ′ . Also, by integrality any
maximal ideal of S must contract to m. Hence, there is a bijection between
the nonmaximal primes of R and those of S.
Suppose S is local. The only possibility of non-bijection of Spec happens
at the maximal ideals, but it is clear that the unique maximal ideal of S
contracts to m. Thus, R → S induces a bijection on Spec even though
R 6= S. Then by the implication (a) =⇒ (b) of Theorem 4.7, R cannot be
perinormal.
On the other hand, if S is not local, then by minimality of the extension,
there is no local integral overring of R = Rm other than R itself. Also, for
any p ∈ Spec R \ {m}, Rp is integrally closed (because it equals SP , where
P contracts to p), so again there is no local integral overring. The same
observation shows that R satisfies (R1 ), since none of the height one primes
of R are maximal. Then by the implication (c) =⇒ (a) of Theorem 4.7, R
is perinormal.

We close this section by presenting an example that shows that Theorem 4.7 and Corollary 4.8 are in some sense best possible.
Example 4.9. The fact that the last two results are false for arbitrary
Noetherian rings can be demonstrated by [Nag62, Appendix, Example 2]
with m = 0. This example consists of a Noetherian normal ring S with
exactly two maximal ideals m1 and m2 where ht m1 = 1 and ht m2 = 2 and
a field k ⊂ S such that the canonical map k → S/mi is an isomorphism for
i = 1, 2. If R = k + J, where J = m1 ∩ m2 , then Nagata shows that S is the
integral closure of R.
We next claim that the set Spec 1 R is in natural bijection with the set X of
height one primes contained in m2 , and that the corresponding localizations
are equal. To see this, let p ∈ Spec 1 R. Then by integrality, there is some
P ∈ Spec 1 S with P ∩ R = p. But all height one primes of S are in m2
except m1 , and m1 ∩ R = J ) p. Thus, P ⊂ m2 . Hence, contraction gives a
surjective map X ։ Spec 1 R. Finally, if P, P ′ ∈ X with P ∩ R = P ′ ∩ R = p,
then SP = Rp = SP ′ (by Corollary 4.5), whence P = P ′ .
Hence R satisfies (R1 ) and the ring Sm2 satisfies going-down over R. However, the ring Sm2 cannot be a localization of R as the maximal ideal of the
former contracts to the maximal ideal of R. Therefore R is not perinormal.
On the other hand, S is a minimal ring extension of R by [PPL06, Theorem
3.3(b)]. Hence there are no local rings strictly between R and R′ = S and
so condition (c) of Theorem 4.7 is satisfied. Thus both Theorem 4.7 and
Corollary 4.8 are false without some assumption on R.

12

NEIL EPSTEIN AND JAY SHAPIRO

5. Gluing points of generalized Krull domains in high
dimension
In this section, we exhibit a method for constructing perinormal domains
out of pre-existing generalized Krull domains, such that the new domains
enjoy an arbitrary degree of branching-like behavior. We explain how to
interpret this construction either in the algebraic context of pullbacks or the
geometric context of gluing points.
We begin with the following result, which may be known, but we include
a proof for the convenience of the reader.
Lemma 5.1. Let R ⊆ S ⊆ T be ring extensions. Let X := {P ∈ Spec S |
P ∩ R is a maximal ideal}. Suppose that the induced map (Spec S \ X) →
(Spec R\Max R) is injective and the inclusion R ⊆ S satisfies INC. If R ⊆ T
satisfies going-down, so does S ⊆ T .
Proof. Let P1 ⊂ P2 be a chain of two prime ideals of S such that there exists
Q2 ∈ Spec T with Q2 ∩ S = P2 . Then setting pj := Pj ∩ R, j = 1, 2, we have
p1 6= p2 (by INC) and Q2 ∩ R = p2 . Then by the going-down hypothesis
on the extension R ⊆ T , there is a prime ideal Q1 of T with Q1 ⊆ Q2 and
Q1 ∩ R = p1 . But then we have P1 ∩ R = p1 = Q1 ∩ R = (Q1 ∩ S) ∩ R, so
by injectivity of the map in question (since p1 is a non-maximal ideal of R),
we have P1 = Q1 ∩ S, completing the proof.

Theorem 5.2. Let S be a semilocal generalized Krull domain and let m1 , . . . , mn
be its maximal ideals. Assume that n ≥ 2, and ht mj ≥ 2 for all 1 ≤ j ≤ n.
Further suppose that the fields S/mi are all isomorphic to the same field k.
For each i = 1, 2, . . . , n fix an isomorphism αi : k → S/mi . Let R be the
pullback in the diagram
R

f

/S
p

g



k

h


/ S/J

Qn

where J := m1 ∩ · · · ∩ mn = j=1 mj , p is the canonical projection, and h is
Qn
the composition of the maps k →
Qn i=1 S/mi (given by λ 7→ (α1 (λ), . . . , αn (λ))
and the isomorphism between i=1 S/mi and S/J (given by the Chinese Remainder Theorem). Then R is local and perinormal. Also, R is globally
perinormal if S is. But R is not integrally closed, because its integral closure is S.
Proof. We first note that it follows from the properties of a pullback that
as h is an injection (resp. p is a surjection), f is an injection (resp. g is
a surjection). Thus we can view R as a subring of S where J = ker g is
a common nonzero ideal of both rings. Then it follows from Corollary 4.6
that R and S have the same field of fractions.
Next we show that S is integral over R (and hence equals the integral
closure of R). To see this, let s ∈ S. Since J is a common ideal of R and S,

PERINORMALITY – A GENERALIZATION OF KRULL DOMAINS

13

we have
k∼
= R/J ֒→ S/J = S/(m1 ∩ . . . ∩ mn ) ∼
=

n
Y

(S/mj ) ∼
= k × · · · × k,

j=1

where the composite map is just the diagonal embedding. Now k × · · · × k
is integral over k, which means that S/J is integral over R/J. In particular,
there is some monic g ∈ (R/J)[X] such that g(s̄) = 0. But then g lifts to a
monic polynomial G ∈ R[X] such that G(s) ∈ J. Say G(s) = j ∈ J. Then
H(X) := G(X) − j is a monic polynomial over R such that H(s) = 0. It
follows that the integral closure of R is S.
Now we claim that R is local. This will follow if we can show that J is
the Jacobson radical of R, since we already have that J is a maximal ideal
of R. To this end, it suffices to show that for each j ∈ J, 1 − j is a unit of
R. If not, then 1 − j ∈ p for some prime ideal of R, so that 1 − j ∈ pS. But
since 1− j is a unit of S (since J is the Jacobson radical of S), it follows that
pS = S, which contradicts the lying over property of the integral extension
R ⊆ S. This contradiction proves the claim.
Before showing that R is perinormal, we collect some observations about
the relationship between Spec R and Spec S. Let P ∈ Spec S and p = P ∩ R.
By integrality P is a nonmaximal ideal of S if and only if p is a nonmaximal
ideal of R. Furthermore in this case by Corollary 4.5, we have Rp = SP ,
whence ht P = ht p. Since we are assuming that no maximal ideal of S has
height 1, each height one prime of S must contract to a height one prime of
R. Moreover by integrality each height one prime of R is lain over by a prime
of S. Thus the Spec map induces a bijection Spec 1 (S) → Spec 1 (R) where
the corresponding localizations coincide. In particular R satisfies (R1 ).
Now let T be an overring of R such that R ⊆ T satisfies going-down.
Case 1: Suppose JT = T . Then S ⊆ T . To see this, let n be a maximal
ideal of T . Since J * n, we have n ∩ R = p ( J. Then there is some
nonmaximal P ∈ Spec S with P ∩ R = p (since S is integral over R), whence
we have Rp = SP . Hence, S ⊆ SP = R
Tp ⊆ Tn . Since n was an arbitrary
maximal ideal of T , it follows that S ⊆ n∈Max T Tn = T .
Next, since the map Spec S → Spec R is injective on non-maximal ideals,
S is integral over R, and R ⊆ T satisfies going-down, it follows from
Lemma 5.1 that the extension S ⊆ T satisfies going-down. Thus, if T is
local or S is globally perinormal, we have that T = SW for some multiplicative subset W of S. On the other hand, for any maximal ideal mi of S, we
have SWQ= T = JT ⊆ mi T = mi SW , so mi ∩ W 6= ∅. Let zi ∈ mi ∩ W , and
let z := ni=1 zi . Note that z ∈ J and that z is a unit in T . Thus by Corollary 4.5, T = SW = (Sz )V = (Rz )V = RV ′ for appropriate multiplicative
sets V and V ′ , so that T is a localization of R.
Case 2: On the other hand if JT 6= T , then by Proposition 3.9, the
∼
map Spec T → Spec R induces a bijection Spec 1 (T ) → Spec 1 (R), and by
Lemma 3.7 the corresponding localizations are equal. Since we have a similar

14

NEIL EPSTEIN AND JAY SHAPIRO

bijection between Spec 1 (S) and Spec 1 (R), we get
\
\
R⊆T ⊆
TP =
SQ = S.
P ∈Spec 1 (T )

Q∈Spec 1 (S)

where the last equality follows from S being a generalized Krull domain. It
follows that T is integral over R and that J is a common ideal to R, S, and
T , so we have
k∼
= R/J ⊆ T /J ⊆ S/J ∼
= k × · · · × k.
Thus, T /J must be isomorphic to a product of some finite number of copies
of k. But by Lemma 4.2, T is local. Hence, T /J is local as well. Therefore,
T /J ∼

= k, whence T = R.
Example 5.3. For a geometrically relevant example of the above, let B =
k[X, Y ], let pj = (xj , yj ) ∈ k2 be distinct ordered pairs (points of k2 ) for
1 ≤ j ≤ t, let nj := (X − xj , Y − yj ) (the maximal ideal corresponding to
T
T
pj ), J := tj=1 nj , and A := k + tj=1 nj . Note that J is a maximal ideal
of A. Then by the above theorem, the ring AJ is perinormal (even globally
perinormal!), but it isn’t normal unless t = 1 (since there are t maximal
ideals lying over JAJ in the integral closure of AJ .)
By [Fer03, Théorème 5.1], Spec A can be seen, quite precisely, as the
algebro-geometric result of gluing together the points p1 , . . . , pt of A2k together, and Spec AJ is the (geometric) localization at the resulting singular
point.
6. Global perinormality
Next, we explore the related but quite distinct concept of global perinormality. In particular, for Krull domains, there is a strong and surprising
relationship to the divisor class group. We illustrate with examples from
algebraic number theory.
Proposition 6.1. Let R be a globally perinormal domain, and let W be a
multiplicative subset of R. Then RW is globally perinormal as well.
Proof. Let S be an overring of RW that satisfies going-down. Let Q ∈
Spec S. Then by Lemma 2.2, the map Spec SQ → Spec (RW )Q∩RW is surjective. But (RW )Q∩RW = RQ∩R canonically, so that the map Spec SQ →
Spec RQ∩R is surjective. Since Q ∈ Spec S was arbitrarily chosen, Lemma 2.2
applies again to show that the map R → S satisfies going-down, whence since
R is globally perinormal, S must be a localization of R. That is, S = RV for
some multiplicative subset V of R. But since RW ⊆ S, we have W ⊆ V , so
that V ′ := V RW is a multiplicative subset of RW , and S = RV = (RW )V ′ ,
finishing the proof that RW is globally perinormal.

We next give a result analogous to Proposition 2.4.
Proposition 6.2. Let R be a perinormal domain. Then R is globally perinormal if and only if every flat overring of R is a localization of R.

PERINORMALITY – A GENERALIZATION OF KRULL DOMAINS

15

Proof. Suppose R is globally perinormal. Let S be a flat overring of R.
Then S satisfies going-down over R (by flatness), so by global perinormality,
S = RW for some multiplicative set W ⊆ R.
Conversely, suppose every flat overring of R is a localization of R. Let
S be an overring that satisfies going-down over R. By perinormality and
Proposition 2.4, S is flat over R, whence by assumption S is a localization
of R. Hence, R is globally perinormal.

Proposition 6.3. Let R be a generalized Krull domain and let S be a goingdown overring of R. Then
\
S=
Rp =: R∆ ,
p∈∆

where ∆ := {p ∈ Spec 1 (R) | pS 6= S}.
Proof. For any maximal ideal m of S, Sm is local overring of R such that
R ⊆ Sm satisfies going-down. Hence by Theorem 3.10, Sm is a localization of
R – i.e., Sm = Rm∩R . Now, for every p ∈ ∆, there is some such m ∈ Max S
with pS ⊆ m, whereas when p ∈ Spec 1 (R) \ ∆, there is no such m. Also,
every such Rm∩R is generalized Krull, by [Gil72, Corollary 43.6]. Thus:


\
\
\
\

Rm∩R =
Sm =
S=
(Rm∩R )P 
m∈Max S

m∈Max S

=

\

m∈Max S




\

p∈Spec 1 (R), pS⊆m

m∈Max S



Rp  =

\

P ∈Spec 1 (Rm∩R )

Rp = R∆ .

p∈∆


The next theorem involves the divisor class group; hence we restrict our
attention to Krull domains (rather than generalized Krull domains), which
is where the theory of the divisor class group is most well developed.
Theorem 6.4. Let R be a Krull domain.
(1) If the divisor class group Cl(R) of R is torsion, then R is globally
perinormal.
(2) The converse holds when dim R = 1.
Proof. To prove part (1), it is enough to show that ever flat overring of R is
a localization of R, due to Proposition 6.2 and Theorem 3.10. So let S be a
flat overring of R. Then by Theorem 2.3, Sm = Rm∩R for all maximal ideals
m of S. In particular, S is an intersection of localizations of R at prime
ideals of R. But recall that Heinzer and Roitman prove [HR04, Corollary
2.9] that for a Krull domain with torsion divisor class group, any intersection
of localizations of R at prime ideals is in fact a localization of R. Thus, S
is a localization of R, whence R is globally perinormal.
As for part (2), the following statement was proved independently in
[Dav64, Theorem 2], [Gol64, Corollary (1)], and [GO64, Corollary 2.6]:

16

NEIL EPSTEIN AND JAY SHAPIRO

Let R be a Dedekind domain. Then the class group Cl(R) of
R is torsion if and only if every overring of R is a localization
of R.
But any 1-dimensional Krull domain is a Dedekind domain [Mat86, Theorem
12.5]. Hence, (2) follows.

Example 6.5. The ring of integers OK of any finite algebraic extension of
K of Q is globally perinormal. This is because OK is a Dedekind domain
(hence Krull) with finite (hence torsion) class group (cf. [FT93, Theorem
31]). The result then follows from Theorem 6.4.
Example 6.6. If Rm is globally perinormal for all maximal ideals m, it does
not follow that R is globally perinormal, even when R is a Dedekind domain
finitely generated over a field. To see this, let E be any elliptic curve, with
Weierstraß equation f = 0, considered as an affine curve in A2C . Then as a
group, E = E(C) is analytically isomorphic (as an algebraic group) to C/Λ
for some lattice Λ [Sil86, Corollary VI.5.1.1], which in turn is abstractly
isomorphic (as a group) to R/Z × R/Z. The latter has uncountably many
non-torsion elements (namely, whenever either of the two coordinates is irrational). On the other hand, E(C) is isomorphic to a particular subgroup
(the so-called degree 0 part) of the divisor class group of the Dedekind domain R = C[X, Y ]/(f ) [Sil86, Proposition III.3.4], as the latter is the affine
coordinate ring of E(C). Thus, Cl(R) contains (uncountably many) nontorsion elements, so by Theorem 6.4(2), R is not globally perinormal. But
Rm is a DVR for any m ∈ Max R (since R is a Dedekind domain), so Rm is
globally perinormal.
On page 17, we have constructed a chart tracking many of the dependencies we have discussed so far. Note that none of the arrows are reversible,
and that a crossed-out arrow indicates a specific non-implication.
7. Some subtleties of the non-Noetherian case
As usual, nuances exist for general commutative rings that do not come
up when one assumes all rings involved are Noetherian. We explore some of
these in the current section.
Example 7.1. There is a non-Noetherian one-dimensional local integrally
closed domain that isn’t perinormal. In fact, any integrally closed onedimensional local domain that isn’t a valuation domain will suffice. For
example, let K/F be a purely transcendental field extension, let X be an
analytic indeterminate over K, let V := K[[X]], and then let R = F + XV .
Then R is easily seen to be local with maximal ideal XV and integrally
closed (but not completely integrally closed) in its fraction field K((X)). To
see that R has dimension 1, let p ∈ Spec R with 0 ( p ( XV . Then by
Lemma 4.4, the map Rp → VR\p is an isomorphism, so that VR\p is local
and hence equals either V or K((X)). The former is impossible since every
nonunit of R is a nonunit of V , and the latter means that p = 0, which

PERINORMALITY – A GENERALIZATION OF KRULL DOMAINS

17

Dedekind
Noetherian
normal

OK , where K is an
algebraic number field

Krull, with
\
torsion Cl(R)
globally
perinormal
\

Krull
integrally
closed

generalized
Krull
weakly
normal

perinormal

Prüfer

(R1 )

seminormal
contradicts the assumption. Hence, Spec R = {0, XV }, whence dim R = 1.
But V is a going-down local overring of R that is not a localization.
Example 7.2. There exist non-Prüfer, integrally closed integral domains
(necessarily non-Noetherian) that are perinormal but not generalized Krull.
For a concrete example, let k be a field, x, y indeterminates over k, and
R = k[x, y, xy , xy2 , xy3 , . . .], considered as a subring of k(x, y). If m = xR,
then m is a maximal ideal of R of height two (see below). If P is any other
maximal ideal of R, then x 6∈ P . Thus R ⊆ k[x, y]p , where p = P ∩ k[x, y].
Hence RP = k[x, y]p , and so in particular RP is a Krull domain, whence
perinormal. It is also now clear that R is not Prüfer. On the other hand, it
is known (though apparently not written down) that Rm is a valuation ring.
Specifically, Rm is the valuation ring associated to the valuation ν on k(x, y)
with value group Z × Z (ordered lexicographically), where ν(x) = (0, 1) and
ν(y) = (1, 0). We give a brief outline as to why Rm = V , where V is the
valuation ring of ν.
Clearly R ⊂ V , since y/x ∈ V . Moreover the maximal ideal of V is
generated by x, and so Rm ⊆ V . For the reverse containment we can write
an arbitrary element of V as f /g, where f, g ∈ k[x, y]. Evidently we can
assume that g is not divisible by y. Thus ν(g) = (0, m) for some nonnegative
integer m. We can then write g = xm h(x) + yp(x, y), where h(x) ∈ k[x] and
p(x, y) ∈ k[x, y]. Since ν(f ) ≥ ν(g), v(f ) = (n, t) where either n > 0 or
n = 0 and t ≥ m. In either case one can show that f /g can be written in
the form F/G, where F, G ∈ k[x, y] and G ∈
/ m. Thus f /g ∈ Rm , showing
the two rings are equal.
Hence, R is integrally closed. Finally to complete the example we must
know that R is not a generalized Krull domain. However, m = xR is a

18

NEIL EPSTEIN AND JAY SHAPIRO

T
principal prime ideal of height two. Thus x−1 ∈ ( p∈Spec 1 (R) Rp ) \ R, contradicting the definition of a generalized Krull domain.
Example 7.3. Let V be any rank 1 valuation ring and n ∈ N. Recall
that generalized Krull domains are closed under finite polynomial extension [Gil72, Theorem 43.11(3)]. Thus, V [X1 , . . . , Xn ] is a generalized Krull
domain (since V is obviously generalized Krull), hence perinormal (by Theorem 3.10). This provides a large class of examples of perinormal domains,
of arbitrary Krull dimension, that are neither Krull nor Prüfer, even locally.
8. Questions
We close with an incomplete but intriguing list of questions suitable for
further research on perinormality and global perinormality.
Question 1. Let k be a field, and let X, Y, Z, W be indeterminates over that
field. Is the normal hypersurface R = k[X, Y, Z, W ]/(XW − Y Z) globally
perinormal or not? Note that its divisor class group is well-known to be
infinite cyclic [Fos73, Proposition 14.8]. If “yes”, this answer would mean
that the converse to Theorem 6.4(1) is false in dimension 3. If “no”, this
answer would provide evidence that the converse may be true.
Question 2. Let R be a perinormal domain, K its fraction field, L a (finite?)
extension field of K, and S the integral closure of R in L. Is S perinormal?
In the non-Noetherian case, this question is interesting even when we
further stipulate that R is integrally closed.
Question 3. Let R be an integral domain and X an indeterminate over R.
What can one say about the perinormality of R in relation to the perinormality of R[X]? Does one imply the other?
Question 4. Let R be a Noetherian local domain whose completion R̂ is
also a domain. If R is perinormal, is R̂ perinormal as well? What about the
converse?
Question 5. Is every completely integrally closed domain perinormal?
Acknowledgments
We wish to thank David Dobbs, Tiberiu Dumitrescu, Alan Loper, Karl
Schwede, and Dana Weston for interesting and useful conversations at various stages of the project. We also wish to thank the referee for many useful
comments and improvements, especially for Propositions 2.4 and 6.2, which
are due to the referee.
References
[AB69]
[CS46]

Aldo Andreotti and Enrico Bombieri, Sugli omeomorfismi delle varietà algebriche, Ann. Scuola Norm. Sup. Pisa (3) 23 (1969), 431–450.
Irvin S. Cohen and Abraham Seidenberg, Prime ideals and integral dependence,
Bull. Amer. Math. Soc. 52 (1946), 252–261.

PERINORMALITY – A GENERALIZATION OF KRULL DOMAINS

19

[Dav64] Edward D. Davis, Overrings of commutative rings. II. Integrally closed overrings,
Trans. Amer. Math. Soc. 110 (1964), 196–212.
[Dob73] David E. Dobbs, On going down for simple overrings, Proc. Amer. Math. Soc.
39 (1973), no. 3, 515–519.
[Dob74]
, On going down for simple overrings. II, Comm. Algebra 1 (1974), 439–
458.
[DP76] David E. Dobbs and Ira J. Papick, On going down for simple overrings. III, Proc.
Amer. Math. Soc. 54 (1976), 35–38.
[Fer03] Daniel Ferrand, Conducteur, descente et pincement, Bull. Soc. Math. France 131
(2003), no. 4, 553–585.
[FO70] Daniel Ferrand and Jean-Pierre Olivier, Homomorphisms minimaux d’anneaux,
J. Algebra 16 (1970), 461–471.
[Fon80] Marco Fontana, Topologically defined classes of commutative rings, Ann. Mat.
Pura Appl. (4) 123 (1980), 331–355.
[Fos73] Robert M. Fossum, The divisor class group of a Krull domain, Ergebnisse
der Mathematik und ihrer Grenzgebiete, vol. 74, Springer-Verlag, New YorkHeidelberg, 1973.
[FT93] Albrecht Fröhlich and Martin J. Taylor, Algebraic number theory, Cambridge
Studies in Advanced Mathematics, vol. 27, Cambridge Univ. Press, Cambridge,
1993.
[Gil72] Robert Gilmer, Multiplicative ideal theory, Pure and Applied Mathematics,
no. 12, Dekker, New York, 1972.
[GO64] Robert Gilmer and Jack Ohm, Integral domains with quotient overrings, Math.
Ann. 153 (1964), 97–103.
[Gol64] Oscar Goldman, On a special class of Dedekind domains, Topology 3 (1964),
no. suppl. 1, 113–118.
[GT80] Silvio Greco and Carlo Traverso, On seminormal schemes, Comp. Math. 40
(1980), no. 3, 325–365.
[HR04] William Heinzer and Moshe Roitman, Well-centered overrings of an integral domain, J. Algebra 272 (2004), no. 2, 435–455.
[Mat86] Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced
Mathematics, no. 8, Cambridge Univ. Press, Cambridge, 1986, Translated from
the Japanese by M. Reid.
[Nag62] Masayoshi Nagata, Local rings, Interscience Tracts in Pure and Appl. Math.,
no. 13, Interscience, 1962.
[PPL06] Gabriel Picavet and Martine Picavet-L’Hermitte, About minimal morphisms,
Multiplicative ideal theory in commutative algebra, Springer, New York, 2006,
pp. 369–386.
[Rat80] Louis J. Ratliff, Jr., Notes on three integral dependence theorems, J. Algebra 66
(1980), no. 2, 600–619.
[Ric65] Fred Richman, Generalized quotient rings, Proc. Amer. Math. Soc. 16 (1965),
794–799.
[Sil86] Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986.
[Swa80] Richard G. Swan, On seminormality, J. Algebra 67 (1980), 210–229.
[Vit11] Marie A. Vitulli, Weak normality and seminormality, Commutative algebra:
Noetherian and non-Noetherian perspectives, Springer-Verlag, 2011, pp. 441–
480.
[Yan83] Hiroshi Yanagihara, Some results on weakly normal ring extensions, J. Math.
Soc. Japan 35 (1983), no. 4, 649–661.

20

NEIL EPSTEIN AND JAY SHAPIRO

Department of Mathematical Sciences, George Mason University, Fairfax,
VA 22030
E-mail address: nepstei2@gmu.edu
Department of Mathematical Sciences, George Mason University, Fairfax,
VA 22030
E-mail address: jshapiro@gmu.edu

