SEMISTAR DEDEKIND DOMAINS
SAID EL BAGHDADI1 , MARCO FONTANA2 , GIAMPAOLO PICOZZA2
1
Department of Mathematics, Faculte des Sciences et Techniques,
P.O. Box 523, Beni Mellal, Morocco
[email protected]
2 Dipartimento
di Matematica, Universita degli Studi \Roma Tre",
Largo San Leonardo Murialdo 1, 00146 Roma
[email protected]
[email protected]
Let D be an integral domain and ? a semistar operation on D. As
a generalization of the notion of Noetherian domains to the semistar setting,
we say that D is a ?{Noetherian domain if it has the ascending chain condition on the set of its quasi{?{ideals. On the other hand, as an extension the
notion of Prufer domain (and of Prufer v{multiplication domain), we say that
D is a Pr
ufer ?{multiplication domain (P?MD, for short) if DM is a valuation domain, for each quasi{?f {maximal ideal M of D. Finally, recalling that
a Dedekind domain is a Noetherian Prufer domain, we dene a ?{Dedekind
domain to be an integral domain which is ?{Noetherian and a P?MD. For the
identity semistar operation d, this denition coincides with that of the usual
Dedekind domains and when the semistar operation is the v{operation, this
notion gives rise to Krull domains. Moreover, Mori domains not strongly Mori
are ?{Dedekind for a suitable spectral semistar operation.
Examples show that ?{Dedekind domains are not necessarily integrally closed
nor one-dimensional, although they mimic various aspects, varying according
to the choice of ?, of the \classical" Dedekind domains. In any case, a ?{
Dedekind domain is an integral domain D having a Krull overring T (canonically associated to D and ?) such that the semistar operation ? is essentially
\univocally associated" to the v{operation on T .
In the present paper, after a preliminary study of ?{Noetherian domains, we
investigate the ?{Dedekind domains. We extend to the ?{Dedekind domains
the main classical results and several characterizations proven for Dedekind
domains. In particular, we obtain a characterization of a ?{Dedekind domain
by a property of decomposition of any semistar ideal into a \semistar product"
of prime ideals. Moreover, we show that an integral domain D is a ?{Dedekind
domain if and only if the Nagata semistar domain Na(D; ?) is a Dedekind
domain. Several applications of the general results are given for special cases
of the semistar operation ?.
Abstract.
1. Introduction and background results
Dedekind domains play a crucial role in classical algebraic number theory and
their study gave a relevant contribution to a rapid development of commutative ring
theory and ideal theory: Noetherian, Krull and Prufer domains arose in the early
stages of these theories, for generalizing dierent aspects of Dedekind domains.
Key words and phrases. Dedekind domain, semistar operation, Krull domain, Mori domain,
Prufer domain, Nagata ring.
2000 Mathematics Subject Classication: 13A15, 13G05, 13E99.
Acknowledgment: Part of this work was done while the rst named author was visiting the
Mathematical Department of Universita degli Studi \Roma Tre", with a visiting grant by INdAM.
The second and the third named authors were supported in part by a research grant by MIUR
2003/2004.
1
2
Star operations provided new insight in multiplicative ideal theory. For instance,
the use of the { and {operations has produced a common treatment and a deeper
understanding of Dedekind and Krull domains. In 1994, Okabe and Matsuda [40]
introduced the semistar operations, extending the notion of star operation and the
related classical theory of ideal systems, based on the pioneering works by W. Krull,
E. Noether, H. Prufer and P. Lorenzen (cf. [35] and [23]). Semistar operations, due
to a major grade of exibility with respect to star operations, provide a natural
and general setting for a wide class of questions and for a deeper and comparative
study of several relevant classes of integral domains (cf. for instance [40], [36], [37],
[12], [13], [14], [15], [11], and [24]).
In this paper, we explore a general theory of Dedekind{type domains, depending
on a semistar operation. A rst attempt in this direction was done by Aubert [4]
and, more extensively, by Halter-Koch [23, Chapter 23], where the author investigated Dedekind domains in the language of nitary ideal systems on commutative
monoids. Our approach is based on the classical multiplicative ideal theory on integral domains, as in Gilmer's book [19], extended in a natural way to the semistar
case. This approach has already produced a generalized and covenient setting for
considering Kronecker function rings ([13], [14], and [15]), Nagata rings [15], and
Prufer multiplication domains [11].
Note that the module systems approach on commutative monoids, developed recently by Halter-Koch in [26], provides an alternative general frame for (re)considering semistar operations on integral domains and related topics. More precisely,
most of the results contained in this paper are of purely multiplicative nature and
remain valid in the more general setting of commutative cancellative monoids (cf.
also Remark 1.2).
Recall that a Dedekind domain is a Noetherian Prufer domain. Let be an
integral domain and a semistar operation on . As a generalization of Noetherian
domains to the semistar setting, we dene to be a {Noetherian domain if it has
the ascending chain condition on the set of the ideals of canonically associated
to (called quasi{ {ideals); equivalently, a {Noetherian domain is a domain in
which each nonzero ideal is f {nite (Lemma 3.3 and Remark 3.6 (1)). For instance,
as we will see later, Noetherian, Mori, and strong Mori domains are examples of
{Noetherian domains, for dierent {operations.
On the other hand, as an extension the notion of Prufer domain (and of Prufer
{multiplication domain), given a semistar operation on an integral domain ,
we say that is a Prufer semistar multiplication domain (P MD, for short) if
is a valuation domain, for each maximal element in the nonempty set of the
ideals of associated to the nite type semistar operation canonically deduced
from (i.e., the quasi{ f {maximal ideal of ). Finally, we dene a {Dedekind
domain ( {DD, for short) to be an integral domain which is {Noetherian and a
P MD. For the identity semistar operation , this denition coincides with that of
the usual Dedekind domains and when the semistar operation is the {operation,
this notion gives rise to Krull domains. Moreover, Mori domains not strongly Mori
are {Dedekind for a suitable spectral semistar operation (Example 4.22).
In the general semistar setting, {Dedekind domains are not necessarily integrally closed nor one-dimensional, although they mimic various aspects, varying
according to the choice of , of the \classical" Dedekind domains. In any case, a
{Dedekind domain is an integral domain having a Krull overring (canonically
associated to and ) such that the semistar operation is essentially \univocally
associated" to the {operation on (Remark 4.21).
In the present paper we develop a theory which enlightens dierent facets of the
{Dedekind domains and shows the interest in studying these new classes of integral
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SEMISTAR DEDEKIND DOMAINS
3
domains of Dedekind-type, parametrized by semistar operations. After recalling in
the present section the main data needed for this work, in Section 2, as a rst
step to the main goal, we introduce and study the concept of \semistar almost
Dedekind domains" ( {ADD, for short), which provides a natural generalization of
the classical notion of almost Dedekind domains. Our study, in the particular case
of = , extends and completes the investigation on {almost Dedekind domains
initiated by Kang [32, Section IV]. Among the main results proven in this section,
we have that an integral domain is a {ADD if and only if the Nagata semistar
domain Na( ) is an almost Dedekind domain (in particular, in this case, Na( )
coincides with the Kronecker semistar function ring Kr( )).
Section 3 is devoted to the study of the semistar Noetherian domains. In particular, we investigate the local{global behaviour of this notion and we obtain several
relevant results on {Noetherian domains, in case of stable semistar operations.
In Section 4, we introduce and study the semistar Dedekind domains. We extend
to the {Dedekind domains the main classical results and several characterizations
proven for Dedekind domains. In particular, we obtain a characterization of a
{Dedekind domain by a property of decomposition of any semistar ideal into a
\semistar product" of prime ideals. Moreover, we show that an integral domain
is a {DD if and only if the Nagata semistar domain Na( ) is a Dedekind
domain (in particular, in this case, Na( ) coincides with the Kronecker semistar
function ring Kr( ), which is in fact a PID).
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Let be an integral domain with quotient eld . Let F ( ) denote the set
of all nonzero {submodules of and let F ( ) represent the nonzero fractional
ideals of (i.e., F ( ) := f 2 F ( ) j for some nonzero element 2 g).
Finally, let f ( ) be the set of all nonzero nitely generated -submodules of (it
is clear that f ( ) F ( ) F ( )).
A map : F ( ) ! F ( ) 7! , is called a semistar operation on if, for
all 2 , 6= 0, and for all
2 F ( ), the following properties hold:
( 1) ( ) = ;
( 2) implies ;
( 3) and := ( ) = ;
cf. for instance [40], [37], [36] and [12]. Recall that, given a semistar operation on
, for all
2 F ( ), the following basic formulas follow easily from the axioms:
( ) =( ) =( ) =( ) ;
( + ) =( + ) =( + ) =( + ) ;
( : ) ( : ) = ( : ) = ( : ) if ( : ) 6= 0;
( \ ) \ = ( \ ) if \ 6= (0);
cf. for instance [12, Theorem 1.2 and p. 174].
A (semi)star operation on an integral domain is a semistar operation, that
restricted to the set F ( ) of fractional ideals, is a star operation on [19, (32.1)].
It is very easy to see that a semistar operation on is a (semi)star operation (on
) if and only if = .
Examples 1.1. (1) The rst (trivial) examples of semistar operations are given
by (or, simply, ), called the identity (semi)star operation on , dened by
D := , for each
2 F ( ) and by (or, simply, ), dened by D := , for
each 2 F ( ).
More generally, if is an overring of , we can dene a semistar operation on ,
denoted by f g and dened by T := , for each 2 F ( ). It is obvious
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that = f g, = f g and that f g is a semistar non{(semi)star operation
on if and only if ( .
(2) If is a semistar operation on , then we can consider a map f : F ( ) !
F ( ) dened, for each 2 FS( ), as follows:
f :=
f j 2 f ( ) and g.
It is easy to see that f is a semistar operation on , called the semistar operation
of nite type associated to . Note that, for each 2 f ( ),
= f . A semistar
operation is called a semistar operation of nite type if = f . It is easy to see
that ( f)f = f (that is, f is of nite type).
For instance, if (or, simply, ) is the {(semi)star operation on dened by
:= ( 1) 1 for each 2 F ( ) with 1 := ( :K ) := f 2 j g ,
then the semistar operation of nite type ( ) (or, simply, f ) associated to is
denoted by (or, simply, ) and it is called the {(semi)star operation on (note
that = = ).
Note also that, for each overring of , the semistar operation f g on is a
semistar operation of nite type.
7! := Tf
j 2
(3) If Spec( ), the map : F ( ) ! F ( ),
g, is a semistar operation on [12, Lemma 4.1]. A semistar operation is called
a spectral semistar operation on if there exists a subset of Spec( ) such that
= . If = f g, then f g is the spectral semistar operation on dened by
P :=
, for each 2 F ( ), i.e. f g = f P g. If = ;, then ; = .
We say that a semistar operation is stable (with respect to nite intersections) if
( \ ) = \ , for each
2 F ( ). For a spectral semistar operation the
following properties hold [12, Lemma 4.1]:
= .
(3.a) For each
2 F ( ) and for each 2 ,
(3.b) The semistar operation is stable.
(3.c) For each
2 , \ = .
(3.d) For each nonzero integral ideal of
such that \ 6= , there exists a
prime ideal 2 such that .
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If 1 and 2 are two semistar operations on , we set 1 2, if 2 1 2 ,
for each1 2 F ( ). It is easy to see that 1 2 if and only if ( 1 ) = 2 =
( 2 ) . Obviously, if 1 2, then ( 1)f ( 2)f ; moreover, for each semistar
operation on , we have f . In particular, ; furthermore,
it is not diÆcult to see that, for each (semi)star operation on , we have and f [19, Theorem 34.1(4)].
A quasi{ {ideal of is a nonzero ideal of such that = \ . This notion
generalizes the notion of {ideal for a star operation on , which is a nonzero ideal
of such that = . More precisely, it is clear that, for a (semi)star operation
, the quasi{ {ideals coincide with the {ideals.
Note that each nonzero ideal of , such that ( , is contained in a (non
trivial) quasi{ {ideal of : in fact, the ideal \ is a quasi{ {ideal of and
\ .
A quasi{ {prime of is a nonzero prime ideal of that is also a quasi{ {ideal
of . A quasi{ {maximal ideal of is a (proper) ideal of , which is maximal in
the set of all quasi{ {ideals of .
If is a semistar operation of nite type on , with 6= , each quasi{ {ideal of
is contained in a quasi{ {maximal ideal. Moreover, each quasi{ {maximal ideal
of is prime [12, Lemma 4.20]. We denote by M( ) the set of all quasi{ {maximal
ideals of . Thus, if = f and is not a eld, then M( ) 6= ;.
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SEMISTAR DEDEKIND DOMAINS
5
Examples 1.1. (4) If is a semistar operation on , we denote by ~ the spectral
semistar operation M( f ), induced by the set M( f ) of the quasi{ f {maximal ideals
of , i.e. for each 2 F ( ~): T
:= f
j 2 M( f )g .
The semistar operation ~ is stable and of nite type and ~ f (cf. [13] and
also [12, p. 181] for an equivalent denition of ~; see [28], [20], [2, Section 2] for
an analogous construction in the star setting). Note that, when is a (semi)star
operation on , then also ~ is a (semi)star operation on .
If = , then obviously ~ = . If = , then ~ is the (semi)star operation that
we denote by (or, simply, ), following Wang Fanggui and R.L. McCasland (cf.
[44], [45] and [43]). Note that, for = M( f ), the semistar operation ~ satises
the properties (3.a){(3.d), stated above for a general spectral semistar operation.
(5) Let
be an integral domain and an overring of . Let be a semistar
operation on . We can dene a semistar operation _ T : F ( ) ! F ( ) on , by
setting:
_ T :=
for each 2 F ( )( F ( )) ;
[13,
Proposition
2.8].
When
=
, we denote simply by _ the (semi)star operation
_ D? on . T
Note that (_f ) = (_ T)f [11, Lemma 3.1]. In particular, if = f then _ T is a
semistar operation of nite type on .
(6) On the other hand, if is a semistar operation on an overring of an integral
domain , we can construct a semistar operation . D : F ( ) ! F ( ) on , by
setting:
. D := ( ) for each 2 F ( )
[13, Proposition 2.9].
For more details on the semistar operations considered in (5) and (6), cf. [40] and
[13].
Remark 1.2. Let be a semistar operation on an integral domain . For each
nonzero ideal of , dene ( ) := \ . Then it is easy to see that the map
7! ( ) denes a weak ideal system (= {system in the sense of K. E. Aubert)
on (as a commutative cancellative monoid, disregarding the additive structure),
cf. [23, Chapter 2], therefore the theory developed in [23, Part A] applies. In
particular, ( f ) = ( ) [23, page 25], M( ) = ( ){max( ) [23, page 57], and
e = ( ) [ ] [25, Denition 3.1 and Proposition 3.2].
Furthermore, using the more general setting of module systems on monoids, the
spectral semistar operations (Example 1.1 (3)) and the semistar operations _ T and
. D , dened in Example 1.1 (5) and (6), have a natural correspondent interpretation
in terms of module systems, which is described in [26], and so the theory developed
in this paper also applies.
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Proposition 1.3.
(1)
Let
D
be an integral domain and
T
an overring of
D
.
Let be a semistar operation on . Denote simply by ? the semistar operation
. D , dened on , then the semistar operations ?_ T and (both dened on
T
D
(2)
) coincide.
Let ? be a semistar operation on D. Denote simply by the semistar operation
_ T, dened on T , then ? . D (note that both semistar operations are dened
?
on D). Furthermore, if T = D? then ? = . D .
T
(1) and the rst statement in (2) are already in [13, Corollary 2.10], [40,
LemmaT 45]. For the last statement note that, for each 2 F ( ), . D = ( ) =
( )_ = ( ) = ( ) = ( ) = .
Proof.
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If is a ring and an indeterminate over , then the ring ( ) := f j 2
[ ] and c( ) = g (where c( ) is the content of the polynomial ) is called
the Nagata ring of [19, Proposition 33.1]. A more general construction of a
Nagata ring associated to a semistar operation dened on an integral domain
was considered in [15] (cf. also [32] and [23, Chapter 20, Ex. 4], for the star case).
Proposition 1.4. [15, Proposition 3.1, Proposition 3.4, Corollary 3.5, Theorem
3.8]. Let be a semistar operation on an integral domain and set ( ) :=
( ) := f 2 [ ] j 6= 0 and (c( )) = g. Let ~~ (respectively, ~_ ) be the
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spectral semistar operation dened on D (respectively, D? ), introduced in Example
1.1((4) and (5)). Then:
(1) NS (?) is a saturated multiplicative subset of D[X ] and N (?) = N (?f ) = D[X ]r
fQ[X ] j Q 2 M(?f )g.
Set Na(D; ?) := D[X ]N (?) and call this integral domain the Nagata ring of D with
respect to the semistar operation .
(2) Max(Na( )) = f [ ] ( ) j 2 M( f )g and M( f ) coincides with the
canonical image in Spec( ) of Max(Na(
)) .
(3) Na( ) = Tf ( ) j 2 M( f )g .
(4) ~ = Na( ) \ for each 2 F ( )
(5) M( ) = M(~)
(6) Na( ) = Na( ~) = Na( ~ ~_ ) ~( )
2
It is clear that Na( ) = Na( f ) and, when = (the identity (semi)star
operation) on , then Na( ) = ( ).
Given a semistar operation on an integral domain , we say that is an e.a.b.
(endlich arithmetisch brauchbar) semistar operation of if, for all
2 f ( ),
( ) ( ) implies that [13, Denition 2.3 and Lemma 2.7].
We recall next the denition of two relevant semistar operations, associated to
a given semistar operation.
Examples 1.1. (7) Given a semistar operation on an integral domain , we
call the semistar integral closure [ ] of , the semistar operation on dened by
setting: [ ]
:= [f(( : ) ) f j 2 f ( )g for each 2 f ( )
and
[ ] := [f [ ] j 2 f ( ) g for each 2 F ( )
It is not diÆcult to see that the operation [ ] is a semistar operation of nite type
on , that [ ] , hence [ ] , and that [ ] is integrally closed [13,
Denition 4.2, Proposition 4.3 and Proposition 4.5 (3)]. Therefore, it is obvious
that if = [ ] then is integrally closed.
(8) Given an arbitrary semistar operation
on an integral domain , it is
possible to associate to , an e.a.b. semistar operation of nite type on ,
called the e.a.b. semistar operation associated to , dened as follows:
a := [f((
) : ) j 2 f ( )g for each 2 f ( )
and
a := [f
a j
2 f ( )g for each 2 F ( )
[13, Denition 4.4]. Note that [ ] , that [ ] = a and if is an e.a.b.
semistar operation of nite type then = [13, Proposition 4.5].
More information about the semistar operations [ ] and can be found in [35],
[39], [40], [22], [23] , [24], and [14].
Let be a semistar operation on and let be a valuation overring of . We
say that is a {valuation overring of if, for each 2 f ( ) , (or
equivalently, f f g (Example 1.1(1)).
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V
D
? V
D
?a
?
?
F
D
F
D
F
?
FV
SEMISTAR DEDEKIND DOMAINS
7
Note that a valuation overring of is a {valuation overring of if and only
if f = . (The \only if" part is obvious; for the \if" part recall that, for
each 2 f ( ), there exists a nonzero element 2 such that = , thus
( ) f = ( ) f = f = = ).
More details on semistar valuation overrings can be found in [14], [15] (cf. also [35],
[22] and [24]).
We recall next the construction of the Kronecker function ring with respect to
a semistar operation (the star case is studied in detail in [19, Section 32] and [23,
Chapter 20, Ex. 6]).
Proposition 1.5. [13, Proposition 3.3, Theorem 3.11, Theorem 5.1, Corollary 5.2,
Corollary 5.3], [14, Theorem 3.5]. Let be any semistar operation dened on an
integral domain with quotient eld and let be the e.a.b. semistar operation
associated to (Example 1.1(8)). Consider the e.a.b. (semi)star operation _ :=
_ D?a (dened in Example 1.1(5)) on the integrally closed integral domain a =
V
V
?
F
F
?
D
?
D
V
D
x
?
FV
?
xV
xV
?
xV
K
FV
xV
FV
?
D
K
?a
?
?a
?a
D
[ ] (Example 1.1((7) and (8))). Set
D
?
?
Kr(
) := f
D; ?
2 D[X ] n f0g and there exists h 2 D[X ] n f0g
such that (c(f )c(h))? (c(g)c(h))? g [ f0g :
j
f =g
f; g
Then we have:
(1)
(2)
(3)
(4)
(5)
(6)
Kr( ) is a Bezout domain with quotient eld ( ) called the Kronecker
function ring of with respect to the semistar operation
Na( ) Kr( )
Kr( ) = Kr( ) = Kr( a _ ) .
a = Kr(
)\
for each 2 F ( ) .
Kr( ) = Tf ( ) j is a {valuation overring of g
If := ( 0 1
) 2 f ( ) and ( ) := 0 + 1 + +
2 [ ]
then:
Kr( ) = ( )Kr( ) = c( )Kr( )
If is an integral domain and is a semistar operation on , we say that a
nonzero ideal is {invertible, if ( 1) = . We dene to be a P MD if each
nonzero nitely generated ideal of is f {invertible (cf. [11] and also [21], [38],
[30], [32], [18]). In particular, note that, if is a star operation, then is a P MD
if and only if D is {Prufer in the sense of [23, Chapter 17].
By using Na( ) and Kr( ), we have the following characterization of a
P MD.
Proposition 1.6. [11, Theorem 3.1, Remark 3.1] Let
be an integral domain and
D; ?
K X
;
D
E
?:
D; ?
D; ? :
D; ?
D; ?a
?
E
D; ?
D; ?
F
V
D
; ?a
K ;
X
E
V
a ; a ; : : : ; an
F
?
D
D
D; ?
D
?
f X
f X
D
a
D; ?
a X
f
?
II
an X
n
K X ;
D; ? :
?
I
:
:::
D
?
D
D
?
D
?
?
?
D
?
?
D; ?
D; ?
?
D
a semistar operation on D. The following are equivalent:
(i) D is a P?MD;
(ii) DQ is a valuation domain, for each Q 2 M(?f );
(iii) Na(D; ?) is a Prufer domain;
(iv) Na(D; ?) = Kr(D; ?);
(v) ~? is an e.a.b. semistar operation;
(vi) ?f is stable and e.a.b.
In particular, D is a P?MD if and only if it is a P~?MD. Moreover, in a P?MD,
~? = ?f .
2
?
Let be an integral domain and an overring of . Let be a semistar
operation on and 0 a semistar operation on . Then, we say that is ( 0){
linked to , if
= )( ) =
D
T
D
D
?
T
D
F
?
D
?
FT
?
0
?
T
T
0
?
;
?; ?
8
for each nonzero nitely generated ideal of . Finally, recall that we say that
is ( 0){at 0 over if it is ( 0){linked to and, in addition, \ = , for
each quasi{ f {maximal ideal of . More details on these notions can be found
in [9] (cf. also [34] and [27]).
2. Semistar almost Dedekind Domains
Let be an integral domain and a semistar operation on . We say that
is a semistar almost Dedekind domain (for short, a {ADD ) if is a rank-one
discrete valuation domain (for short, DVR), for each quasi{ f {maximal ideal of
.
Note that, by denition, {ADD = f {ADD and that, if = (= the identity
(semi)star operation), we obtain the classical notion of \almost Dedekind domain"
(for short, ADD) as in [19, Section 36]. Note that, If = , the {ADDs coincide
with the {almost Dedekind domains studied by Kang [32, Section 4]; more generally, if is a star operation, then is a {ADD if and only if is a {almost
Dedekind domain in the sense of [23, Chapter 23]. Note also that, a eld has only
the identity (semi)star operation and thus a eld is, by convention, a trivial example of a ( {)ADD (since, in this case, M( ) = ;).
An analogous notion of generalized almost Dedekind domain was considered in the
language of ideal systems on commutative monoids in [23, Chapter 23].
Remark 2.1. Let 1 2 be two semistar operations on
such that ( 1)f ( 2)f .
If is a 1{ADD, then is a 2{ADD. In particular:
{ is a ADD ) is a {ADD, for each semistar operation on ;
{ if is a (semi)star operation on (so, ), then:
is a {ADD ) is a {ADD (and, hence, is integrally closed).
Note that, in general, for a semistar operation , a {ADD may be not integrally
closed. For instance, let be a eld and := [ ] = + , where :=
is the maximal ideal of the discrete valuation domain . Set := + , where
is a non integrally closed integral domain with quotient eld (hence, is not
integrally closed [10, Proposition 2.2(10)]). Take := f g on . Then, we have
= f , _ = is the identity (semi)star operation on and M( ) = f g (by
[15, Lemma 2.3(3)]) and = [10, Proposition 1.9]. So is a {ADD which is
not integrally closed (hence, in particular, is not an ADD).
F
?; ?
D
D
?; ?
?
T
D
Q
DQ D
TQ
T
D
?
D
?
D
DM
?
M
D
?
?
?
?
d
v
v
t
?
D
?
d
D
d
? ;?
D
?
D
D
D
D
?
?
D
D
?
?
D
?
v
D
K
T
?
K X
K
M
T
M
D
R
R
K
?
T
?
? T
dT
XT
M
D
D
T
DM
D
v
?
?
?
?
?
?
?
?f
T
D
M
?
D
Let D be an integral domain, which is not a eld, and ? a semistar operation on D. Then:
(1) D is a ?{ADD if and only if DP is a DVR, for each quasi{?f {prime ideal P
of D.
(2) If D is a ?{ADD, then D is a P?MD and each quasi{?f {prime of D is a
quasi{?f {maximal of D.
(3) Let T be an overring
of D and ?0 a semistar operation on T . Assume that
0
0
D T is a (?; ? ){linked extension. If D is a ?{ADD, then T is a ? {ADD.
?
(4) If D is a ?{ADD, then D is a ?_ {ADD.
Proposition 2.2.
(1) It follows easily from the fact that each quasi{ f {prime is contained in
a quasi{ f {maximal [15, Lemma 2.3(1)].
(2) is a straightforward
consequence of (1) and of Proposition 1.6 ((i),(ii)).
(3) Let 2 M( f0 ), then ( \ ) f 6= [9, Proposition 3.2]. Let \ be
a quasi{ f {maximal ideal of . We have \ . So = \ =
, because
is a DVR (by assumption is a {ADD). From this proof we
deduce
also
that
\ (= ) is a quasi{ f {maximal ideal of , for each quasi{
0 {maximal ideal of .
f
(4) It follows from [9, Lemma 3.1(e)] and (3).
Proof.
?
?
N
?
N
?
DM
D
?
D
?
M
DM
DM
D
N
?
D
N
D
T
M
?
DN
D
TN
TN
?
D
N
DN
D
D
SEMISTAR DEDEKIND DOMAINS
9
We will show that, for a converse of Proposition 2.2(2), we will
need additional conditions (cf. Theorem 2.14((1),(3), (4)).
(2) The converse of Proposition 2.2(4) is not true in general. Indeed, let
be a
eld and a proper subeld of . Let := [[ ]] and := + , where
:= is the maximal ideal of . Take := f g on . Note that = f and
that _ = is the identity (semi)star operation on . We have that = is a
_ {ADD = ADD (since is a DVR), but is not a {ADD, since is a quasi{
and (by [10, Proposition 1.9]) = is not a valuation
f {maximal ideal of
domain.
Remark 2.3. (1)
K
k
M
K
K
XT
T
T
T
?
K
?
X
? T
dT
D
?
T
?
M
D
?
T
T
k
D
T
?
D
?
D
?
M
DM
D
Let D be an integral domain and ? a (semi)star operation on D.
Then the following are equivalent:
(1) D is a ?{ADD.
(2) D is a t-ADD and ?f = t.
Proposition 2.4.
Proof. (1) ) (2) By Remark 2.1, if is a {ADD, then is a {ADD or, equivalently, a {ADD. Moreover, by Proposition 2.2(2) and [11, Proposition 3.4], = .
The converse is clear.
D
?
D
v
t
?f
t
Let D be an integral domain, which is not a eld, and ? a semistar
operation on D. The following are equivalent:
(1) D is a ?{ADD.
(2) Na(D; ?) is an ADD (i.e. Na(D; ?) is a 1-dimensional Prufer domain and
contains no idempotent maximal ideals).
(3) Na(D; ?) = Kr(D; ?) is an ADD and a Bezout domain.
Theorem 2.5.
(1) , (2). By Proposition 1.4(2), the maximal ideals of Na( ) are of
the form Na( ), where 2 M( f ). Also, for each 2 M( f ), we have
Na( ) Na( ) = ( ). Moreover, it is well-known that, for 2 M( f ),
is a DVR if and only if ( ) is a DVR [19, Theorem 19.16 (c), Proposition
33.1 and Theorem 33.4 ((1),(3))]. From these facts we conclude easily.
(1))(3). If is a {ADD, in particular is a P MD (Proposition 2.2(2)), then
Na( ) = Kr( ), by Proposition 1.6 ((i ),(iv)). Therefore, we deduce that
Na( ) is a Bezout domain (Proposition 1.5(1)) and an ADD by (1))(2).
(3) ) (2) is trivial.
Proof.
D; ?
M
D; ?
D; ? M
M
?
M
DM X
D;?
DM
?
M
?
DM X
D
D; ?
?
D
?
D; ?
D; ?
Let D be an integral domain and
The following are equivalent:
(1) D is a ?{ADD.
(2) D is a ~?{ADD.
(3) D~? is a ~?_ {ADD.
(4) D~? is a t{ADD and ~?_ = tD~? .
Corollary 2.6.
?
a semistar operation on
D
.
Note that Na( ) = Na( ~) = Na( ~ ~_ ) (Proposition 1.4(6)), then apply Theorem 2.5((1),(2)) to obtain the equivalences (1) , (2) , (3). The equivalence (3),(4) follows from Proposition 2.4.
Next goal is a characterization of {ADD's in terms of valuation overrings, in the
style of [19, Theorem 36.2]. For this purpose, we prove preliminarly the following:
Proof.
D; ?
D; ?
?
D ;?
?
Let D be an integral domain and ? a semistar operation on
be a valuation overring of D. Then the following are equivalent:
(1) V is a ~?{valuation overring of D.
(2) V is (~?; dV ){linked to D.
Lemma 2.7.
V
D
. Let
10
(1) ) (2): Since is a ~{valuation overring, then ~ f g. Thus, the
present implication follows from the fact that _ f g = (so ~_ = ) and from
[9, Lemma 3.1(e)].
(2) ) (1):
Let be the maximal ideal of (which is (~ ){linked to ). Then
( \ )~ 6= ~ by [9, Proposition 3.2 ((i))(v)) ]. Thus, there exists 2 M( f ) =
M(~) (Proposition 1.4(5)) such that \ . Hence
\ . So,
if 2 f ( ), then ~ . Therefore, is a ~{valuation overring of
.
Proof.
V
?
?
V
? V
N
N
?
D
D
dV
V
?
dV
?; dV
D
?
M
?
N
F
? V
V
D
F
?
F DM
D
M
DM
FV
V
DN
?
V
D
?
D
Let D be an integral domain, which is not a eld, and ? a semistar
operation on D. The following are equivalent:
(1) D~?is ?{ADD.
(2) D is integrally closed and each ~?{valuation overring of D is a DVR.
(3) D~? is integrally closed and each valuation overring V of D, which is (~?; dV ){
linked to D, is a DVR.
(4) D~? is integrally closed and each valuation overring V of D, which is (?; dV ){
linked to D, is a DVR.
Theorem 2.8.
(1) ) (2). Since ~ = Tf j 2 M( f )g and is a DVR, for each
2 M( f ), then ~ is integrally closed. Now, let be a ~{valuation overring of
, then for some 2 M( f ) [15, Theorem 3.9]. Since is a DVR,
then = (is a DVR).
(2) , (3). Follows immediately from Lemma 2.7.
(3) ) (4). It is an immediate consequence of the fact that ~ (cf. [9, Lemma
3.1(h)]).
(4) ) (1). Let 2 M( f ) and be valuation overring of . Then = r
is ( ){linked to (cf. [9, Example 3.4(1)]).~ Hence, by assumption, ~ is a DVR.
Furthermore, is integrally closed, since and thus =
M \ ~? .
So is an ADD, by [19, Theorem 36.2], that is, is a DVR. Therefore is a
{ADD.
Proof.
D
M
?
D
V
V
D
?
DM
M
?
?
DM
V
DM
M
?
?
DM
DM
?
M
?; dV
?
V
?
DM
V
D
VD
M
V
DM
D
?
DM
DM
?
DM
D MD
DM
D
D
?
Let D be an integral domain, which is not a eld. Then the following are equivalent:
(1) D is t{almost Dedekind domain.
(2) D is integrally closed and each w{valuation overring of D is a DVR.
(3) D is integrally closed and each t{linked valuation overring of D is a DVR.
Corollary 2.9.
This is an immediate consequence of Theorem 2.8 and of the wellknown fact
that for a valuation domain , = = (cf. also [9, Section 3] for the
{linkedness).
Remark 2.10. If
is a {ADD, which is not a eld, then, by Theorem 2.8 and
by the fact that a {valuation overring is a ~{valuation overring, each~ {valuation
overring of is a DVR. Note that the converse is not true, even if is integrally
closed. Let and be as in Remark 2.3(2). Assume that is algebraically~
closed in . Since = f g, then = f , M( f ) = f g and =
=
is integrally closed, where ~ = . Moreover, each {valuation overring of is
necessarily a valuation overring of (since = f = = f = ). This
implies that each {valuation overring of is a DVR (since the only non trivial
valuation overring of is , which is a DVR). Therefore, by Proposition 1.4(6)
) = ( ) ( Kr( ) = Kr( ) =
and 1.5(5), Na( ) = Na( ~ ~_ ) = Na(
( ) (where is an indeterminate over and ). On the other hand, since
t.deg ( ) 1, it is possible to nd (~{) valuation overrings of (of rank 2)
contained in [19, Theorem 20.7].
Proof.
V
dV
wV
tV
t
D
?
?
?
?
D
D
D
T
K
?
k
? T
?
?
?
?
D ;?
D; dD
T
?
T
DM
D
?
D
D
D
?
V
V
?
V
?
T
Z
K
D
D
T
k
M
?
T
?
T Z
?
dD
T
D; ?
?
D Z
D; ?
D
D
T ; dT
?
SEMISTAR DEDEKIND DOMAINS
11
Let be an integral domain and a semistar operation on . For each quasi-prime of , we dene the -height of (for short, -ht( )) the supremum
of the lengths of the chains of quasi{ {prime ideals of , between prime ideal (0)
(included) and . Obviously, if = is the identity (semi)star operation on ,
then -ht( ) = ht( ), for each prime ideal of . If the set of quasi{ {primes of
is not empty, the -dimension of is dened as follows:
-dim( ) := Supf -ht( ) j is a quasi{ {prime of g .
If the set of quasi{ {primes of is empty, then we pose -dim( ) := 0.
Note that, if 1 2, then 2-dim( ) 1-dim( ). In particular, -dim( ) dim( ) = dim( ) (= Krull dimension of ), for each semistar operation on
. Note that, recently, the notions of -dimension and of -dimension have been
received a considerable interest by several authors (cf. for instance, [29], [41] and
[42]).
D
?
?
P
D
D
?
P
?
?
P
d
?
P
D
d
D
P
D
P
?
?
?
?
?
D
?
D
D
D
P
P
?
D
D
?
?
?
D
?
D
D
D
?
D
d
D
D
?
t
Lemma 2.11.
P
w
Let D be an integral domain and
a semistar operation on D, then
?
~- dim( ) = Supfht( ) j 2 M( f ) = M(~)g =
= Supfht( ) j is a quasi{~{prime of g
Proof. Let
2 M( f ) and be a nonzero prime ideal of . Since M( f ) =
M(~) (Proposition 1.4(5)) we have ~ \ \ = . So is a
quasi{~{prime ideal of . Hence ht( ) = ~-ht( ), so we get the Lemma.
Remark 2.12. Note that, in general,
is a quasi{ f {prime of g .
f -dim( ) Sup fht( ) j
Moreover, it can happen that f -dim( ) Supfht( ) j is a quasi{ f {prime of
g, as shows the following example.
Let be a DVR , with maximal ideal , dominating a two-dimensional local
Noetherian domain , with maximal ideal [8] (or [7, Theorem]), and let :=
f g . Then, clearly, = f and the only quasi{ f {prime ideal of is , since
if is a nonzero prime ideal of , then = = , for some integer 1. Therefore, if we assume that is quasi{ f {ideal of , then we would have
= \ = \ , which implies that = . Therefore, in this
case, 1 = f -dim( ) = f -ht( ) Supfht( ) j is a quasi{ f {prime of g =
ht( ) = dim( ) = 2. Note that, in the present example, ~ coincides with the
identity (semi)star operation on .
It is already known that, when = , it may happen that {dim( ) {dim( ),
[42, Remark 2].
The following lemma generalizes [19, Theorem 23.3, the rst statement in (a)].
?
D
M
M
P
M
?
P
D
D
:
D
D
?
?
?
M
?
?
?
P
?
P
P
M
?
P
D
P DM
P
P
M
P
?
?
D
?
D
D
P
P
?
D
T
N
D
? T
M
?
?
P
D
P
?
P
P
PT
D
N
?
k
D
M
?
?
PT
M
?
M
N
?
k
k
P
P
M
D
k
D
D
M
P
?
D
D
?
D
?
v
t
D
< w
D
Let D be a P?MD. Let Q be a nonzero P {primary ideal of
some prime ideal P of D, and let x 2 D r P . Then Q~? = (Q(Q + xD))~? .
Lemma 2.13.
D
, for
Let 2 M( f ). If 6 , then
= 2 = ( + ) (= ).
If , then
is
-primary and 2 r
; so
= 2 ,
by [19, Theorem 17.3(a)], since ~ is a valuation domain. Thus
=( +
~
) , hence = ( ( + )) .
Let be a semistar operation on an integral domain . We say that has the {
cancellation law (for short, {CL) if
2 F ( ) and ( ) = ( ) implies
that = . The following theorem provides several characterizations of the
semistar almost Dedekind domains and, in particular, it generalizes [19, Theorem
36.5] and [32, Theorem 4.5].
Proof.
M
Q
?
M
QDM
Q
M
QDM
P DM
x
Q DM
DM
Q Q
xD DM
P DM
DM
?
Qx DM
Q
Q Q
xD
B
?
QDM
D
C
QxDM
Q
?
?
?
DM
QDM
A; B; C
D
D
AB
?
AC
?
?
?
Let D be an integral domain which is not a eld and let
semistar operation on D. The following are equivalent:
Theorem 2.14.
?
be a
12
(1) is -ADD.
(2) has the ~{cancellation law.
(3) is a P MD, f - dim( ) = 1 and ( 2) f 6= f , for each 2 M( f ) (=
M(~)).
(4) is a P MD and \ 1( ) f = 0 for each proper quasi{ f {ideal of .
(5) is a P MD and it has the f {cancellation law.
Proof. (1) ) (2). Let
be three nonzero (fractional) ideals of such
that
( )~ = ( )~. Let 2 M( f ). Then, we have
= ( )~ =
( )~ =
(we used twice the fact that ~ is spectral, dened by M( f )).
Moreover, since ~ is~ a DVR then, in particular,
is principal, thus
=
. Hence = .
(2) ) (3). If has ~{CL, then in particular, ~ is an e.a.b. semistar operation on [13, Lemma 2.7], thus is ~a P MD~ (Proposition 1.6 ((v))(i)) ). Let
2 M( f ). Clearly, by ~{CL, ( 2) 6= , and hence ( 2 ) f 6= f (since
~ = f by Proposition 1.6). Next we show that ht( ) = 1, for each 2 M( f ).
Deny, let be~ a nonzero prime ideal of ~ and let 2 r . By Lemma 2.13,
~ = ( ( + )) . Hence ~ = ( + ) , by ~{CL. So + 6 , which is
impossible. Hence ht( ) = 1, for each 2 M( f ). Therefore, we conclude that
f -dim( ) = ~-dim( ) = 1 (Lemma 2.11).
(3) ) (4). Recall that each proper quasi{ f {ideal is contained in a quasi{ f {
maximal ideal, then it suÆces to show that \ 1( ) f = 0, for each 2 M( f ).
Since,
by assumption ( 2) f 6= f , then in particular ( 2)~ 6= ~, and so
2
6=
. Henceforth f
g 1 is the set of
-primary ideals of
[19, Theorem 17.3(b)]. From the assumption we deduce that dim( ) = 1 (because f = ~ by Proposition 1.6), then \ 1
= 0 [19, Theorem 17.3 (c) and
(d)]. In particular, we have \ 1( )~ \ 1 ( )~ = \ 1 (
)=
0, therefore \ 1( ) f = 0.
(4) ) (1). Let 2 M( f ). It is easy to see that ( )~ =
\ ~ , for
each 1. So, (\ 1
) \ ~ \ 1(
\ ~ ) = \ 1 ( )~ f
\ 1( ) = 0 (the last equality holds by assumption). Hence \ 1
= 0,
since
is an essential valuation overring of ~ . It follows that is a DVR
[19, p. 192 and Theorem 17.3(b)].
(2) , (5) is a consequence of the fact that in a P MD, e = f and that the ~{CL
implies P MD.
Remark 2.15. As a comment to Theorem 2.14 ((1),(5)), note that
may have
the f {CL without being a {ADD. It is suÆcient to consider the example in Remark 2.3(2). In that case, = f and ~ = , since M( f ) = f g. Clearly, has
the {cancellation law (because is a DVR), but, as we have already remarked,
is not a {ADD, hence, equivalently, has not the (~{)cancellation law.
Next result provides a generalization to the semistar case of [19, Theorem 36.4
and Proposition 36.6].
D
?
D
?
D
?
?
D
?
M
M
?
M
?
?
D
?
D
?
I
n
n ?
?
I
A; B; C
?
AB
?
AC
?
AC
DM
D
M
?
ABDM
AC DM
B
?
C
?
?
D
?
?
?
?
M
M
?
?
M
?
?
M
P P
D
?
xD
D
?
P
xD
M
?
D
?
M
x
?
M
?
M
?
M
M
P
P
BDM
?
D
?
DM
?
ADM
D
M
?
AB
?
DM
C DM
D
?
?
P
P
xD
M
?
D
?
?
M
n
?
M
M DM
M
M DM
M
n
n ?
M
?
?
M
DM
?
M
?
M DM
n
DM
DM
?
?
n
n
M
M
n
M
n
n ?
n
DM
M
n
n ?
?
n
n ?
M
n
n ?
M
M
DM
M
n
DM
D
?
n
M
n
n ?
DM
M
n
M
D
n
DM
?
D
?
M
DM
D
M
n
n
DM
n
n
?
n ?
DM
DM
?
?
?
?
?
D
?
?
?
?
?
?
dD
?
M
D
T
?
D
D
?
Let D be an integral domain, which is not a eld, and ? a
semistar operation on D. The following are equivalent:
(1) D is a ?{ADD.
p
(2) For each nonzero ~?ideal Inof~? D, such that I ?f 6= D? and I =: P is a prime
ideal of D, then I = (P ) , for some n 1.
(3) ~?- dim(D) = 1 and, for each primary quasi{~?{ideal Q of D, then Q~? = (M n)~? ,
for some M 2 M(?f ) and for some n 1.
Proposition 2.16.
p
(1) ) (2) and (3). Let be a nonzero ideal of with f 6= andp =
is prime. Let be a quasi{ f {maximal ideal of such that . So = Proof.
I
M
?
D
D
I
?
I
D
M
?
I
I
P
P
SEMISTAR DEDEKIND DOMAINS
13
, and hence = , since is a DVR. Thus
=
for some 1.
On the other hand, if 2 M( f ) and 6= , then
= =
. Hence
~ = ( )~ , i.e. ~ = ( )~ . The fact that ~-dim( ) = 1 follows from Theorem
2.14((1))(3)) (since, in the present situation, f = ~).
(2) ) (1). Let 2 M( f ). Let be an ideal of and assume thatpp =
,
for some prime ideal of , . Set ~:= \ ~. We have = and
hence f f . By assumption,
= ( ) , for some 1, hence
=( \ ) =
= ~ = ( )~ =
. It follows from [19,
Proposition 36.6] that is an ADD. Hence is a DVR.
(3) ) (1). We can assume = f , since {ADD and f {ADD coincide. Let
2 M( f ) (= M(~) (Proposition 1.4(5)). Since ~-dim( ) = 1, then ht( ) =
dim( ) = 1 (Lemma 2.11). We can now
proceed and conclude as in the proof
of (2) ) (1). (In this case, we have p = ~ and~ so is a {primary
quasi{~{ideal of . Therefore, by assumption, = ( ) , for some 1.)
M
P
M
DM
N
?
I
M
n ?
I
?
I DM
?
P
N
M
n ?
I DN
?
?
A
P
B
A
?
M
A
?
D
D
D DM
M
B
?
B DM
n
DN
P
n ?
A
A
?
D
n ?
P
DM
P
n
P DM
B
P
n
DM
DM
?
?
n
M
?
B
DM
M
DM
DN
DM
P
?
BDM
n
D
?
M
M
?
?
?
?
?
D
M
DM
A
?
D
M DM
B
?
B
M
n ?
M
n
Note that, if is a {ADD, which is not a eld, then necessarily satises the following conditions (obtained from the statements (2) and (3)
of Proposition 2.16; recall that, in this case, f = ~, by Proposition 2.2(2) and
Proposition 1.6):
p
(2f ) For each nonzero ideal of , such that f 6= and =: is a prime
ideal of , then f = ( ) f , for some 1.
(3f ) f - dim( ) = 1 and, for each primary quasi{ f {ideal of , then f =
( ) f , for some 2 M( f ) and for some 1.
On the other hand, may satisfy either (2f ) or (3f ) without being a {ADD.
It is suÆcient to consider the example in Remark 2.3(2). In that case, = f
and M( f ) = f g. Clearly, since is a local one-dimensional domain (in fact,
~-dim( ) = f -pdim( ) = dim( ) = 1), for each nonzero ideal of , with
f 6=
, then = and f = ( ) f , for some 1, since is a DVR.
But, as we have already remarked, is not a {ADD.
Remark 2.17.
D
?
D
?
I
D
?
I
?
P
D
I
n ?
?
?
D
?
n ?
M
I
P
n
D
M
?
?
Q
D
Q
?
n
D
?
?
?
?
I
D
?
D
M
?
D
?
?
D
I
D
M
I
I
?
M
n ?
D
n
D
T
?
3. Semistar Noetherian domains
Let be an integral domain and a semistar operation on . We say that is
a {Noetherian domain if has the ascending chain condition on quasi{ {ideals.
Note that, if (= ) is the identity (semi)star operation on , the {Noetherian
domains are just the usual Noetherian domains and the notions of {Noetherian
[respectively, {Noetherian] domain and Mori [respectively, strong Mori] domain
coincide [5, Theorem 2.1] [respectively, [44]].
Recall that the concept of star Noetherian domain has already been introduced,
see for instance [1], [46] and [18]. Using ideal systems on commutative monoids, a
similar general notion of noetherianity was considered in [23, Chapter 3].
D
?
?
D
D
D
d
dD
?
D
d
v
w
Lemma 3.1.
(1)
Let
D
be an integral domain.
Let ? ?0 be two semistar operations on D, then D is ?{Noetherian implies
0
D is ? {Noetherian.
In particular:
(1a) A Noetherian domain is a ?{Noetherian domain, for any semistar operation ? on D.
(1b) If ? is a (semi)star operation and if D is a ?{Noetherian domain, then
D is a Mori domain.
14
(2)
Let T be an overring of D and a semistar operation on T . If T is {
Noetherian, then D is .D {Noetherian. In particular, if ? is a semistar opera-
tion on , such that is a _ {Noetherian domain, then is a {Noetherian
domain.
Proof. (1) The rst statement holds because each quasi{ 0{ideal is a quasi{ {ideal.
(1a) and (1b) follow from (1) since, for each semistar operation , and, if
is a (semi)star operation, then .
(2) If we have a chain of quasi{. {ideals f g 1 of that does not stop then,
by considering f( ) g 1, we get a chain of quasi{{ideals of that does not
stop, since two distinct quasi{. {ideals 6= 0 of are such that ( ) 6= ( 0 ).
The second part of the statement follows immediately from the fact that, if we set
:= _ , then . = (Proposition 1.3(2)).
Remark 3.2. The converse of (2) in Lemma 3.1 does not hold in general. For
instance, take , where is a Noetherian domain and is a non-Noetherian
overring of . Let := and := f g. Note
that . = . Then, is {
Noetherian, by (1a) of Lemma 3.1, but = = is not {Noetherian (or,
equivalently, _ {Noetherian), because = = _ = _ and is not Noetherian.
However, if = ~, the last statement of (2) in Lemma 3.1 can be reversed, as we
will see in Proposition 3.4.
D
D
?
?
D
?
?
?
?
?
?
D
T
n
I
?
?
v
In n
In T
d
I
D
IT
I T
?
D
T
D
D
T
dT
?
? T
D
T
?
?
?
?
T
T
dT
D
?
T
?
?
T
?
Let D be an integral domain and let ? be a semistar operation on D.
Then, D is a ?{Noetherian domain if and only if, for each nonzero ideal I of D,
there exists a nitely generated ideal J I of D such that I ? = J ? . Therefore, D is
a ?{Noetherian domain if and only if, for each E 2 F (D), there exists F 2 f (D),
such that F E and F ? = E ? . In particular, if ? is a star operation on D and if
D is a ?{Noetherian then ? is a star operation of nite type on D .
Lemma 3.3.
For the \only if" part, let 1 2 , 1 6= 0, and set 1 := 1 . If = 1 we
are done. Otherwise, it is easy to see that 6 1 \ . Let 2 2 r ( 1 \ ) and
set 2 := ( 1 2) . By iterating this process, we construct a chain f \ g 1
of quasi{ {ideals of . By assumption this chain must stop, i.e., for some 1,
\ = +1 \ , and so = ( \ ) = . So, we conclude by taking := .
Conversely, let f g 1 be a chain of quasi{ {ideals in and set := S 1 .
Let be a nitely generated ideal of such that = , so there exists 1
such that and = = . This implies that the chain of quasi{ {ideals
f g 1 stops (in fact, = = \ , for each ).
Proof.
x
I
x
I
I
I
x ;x
?
Ik
x
I
I
?
?
I
D n
k
?
D
?
Ik
Ik
D
?
I
?
J
?
I
D
D
J
Ik
J
In n
?
D
?
In
In n
J
D
I
D
?
D
x D
D
?
Ik
I
?
?
?
Ik
In
I
Ik
J
?
I
I
n
?
k
?
I
Ik
In
?
?
D
n
k
Let D be an integral domain and let ? be a semistar operation
.
(1) Assume that ? is stable. Then D is ?{Noetherian if and only if D? is ?_ {
Noetherian.
(2) D is ~?{Noetherian if and only if D~? is ~?_ {Noetherian.
Proposition 3.4.
on
D
(1) The \if" part follows from Lemma 3.1(2) and Proposition 1.3(2) (without
using the hypothesis of stability). Conversely, let be a nonzero ideal of and
set := \ . Then, = ( \ ) = \ = . Therefore, by Lemma 3.3
(applied_ to ), we can nd _ 2 f ( ) such that and = . Hence,
( ) = = = = . The conclusion follows from Lemma 3.3 (applied to
, since
and
2 f ( )).
(2) is a straightforward consequence of (1).
Proof.
I
J
I
D
J
?
I
D
FD
D
? ?
?
F
?
FD
J
?
?
I
D
I
?
I
FD
D
I
?
D
?
D
I
F
?
?
J
F
?
J
?
?
?
D
?
Let D be an integral domain and ? a semistar operation on
is ?{Noetherian if and only if D is ?f {Noetherian.
Proposition 3.5.
Then,
D
F
?
D
.
SEMISTAR DEDEKIND DOMAINS
15
The \if" part follows from Lemma 3.1(1), since f . The converse follows
immediately from Lemma 3.3.
Remark 3.6. Let
be an integral domain and a semistar operation on .
(1) Let
2 F ( ), we say that is {nite if there exists 2 f ( ) such that
= . From Lemma 3.3 it follows that if is a {Noetherian domain, then
each nonzero fractional ideal is {nite. The converse does not hold in general [18,
Example 18]. However, when = f , 2 F ( ) is {nite if and only if there
exists 2 f ( ) such that = , with [16, Lemma 2.3] (note that the
star operation case was investigated in [1]). From the previous considerations, from
Lemma 3.3 and from Proposition 3.5, we deduce easily that is a -Noetherian
domain if and only if every nonzero fractional ideal of is f -nite.
(2) Note that:
~{Noetherian ) {Noetherian
because ~ (Lemma 3.1(1)). The converse is not true in general. Indeed, if
:= , then f = and ~ = and we know that {Noetherian (= {Noetherian)
is Mori and that {Noetherian is strong Mori [44, Section 4]. Since it is possible
to give examples of Mori domains that are not strong Mori [45, Corollary 1.11], we
deduce that {Noetherian does not imply ~{Noetherian.
In the next result, we provide a suÆcient condition for the transfer of the semistar
Noetherianity to overrings.
Proof.
?
D
E
E
?
F
?
?
D
E
D
?
?
D
F
D
D
?
?
?
?
F
D
E
?
?
F
D
E
?
F
?
E
D
?
?
?
?
?
;
?
v
?
t
?
w
v
t
w
?
?
Let D be an integral domain and let T be an overring of D. Let
be a semistar operation on D and ?0 a semistar operation on T . Assume that T
is (?; ?0){at over D. If D is ~?{Noetherian, then T is ?e0{Noetherian.
Proposition 3.7.
?
Let be a nonzero ideal of . Let 2 M( e0) = M( 0 ) (Proposition
e
1.4(5)). From the0 ( 0){atness, it follows that
=
=
\ . Then,
\f
j 2 M( )g = \f \ j 2 M( 0 )g. Now, \ is a prime of
such that ( \ 0)~ 6= ~ (by [9, Proposition 3.2], since is ( 0){linked to , by
denition of ( ){atness). Hence, \ is a quasi{~{ideal. Consider the ideal
\ of . Since is e{Noetherian, it follows by Lemma 3.3 that there exists a
nitely generated ideal of , such that ~ \ and ~~ = ( \ )~ . Then,
=
\ =( \ ) \ =( \ )
\ =
\ =
\ =
( ) . Thus, ~ = ( ) ~ , with nitely generated ideal of , such that
. Hence, is e0{Noetherian.
Let be an integral domain and a semistar operation on . We say that
has the {nite character property (for short, {FC property) if each nonzero
element of belongs to only nitely many quasi{ {maximal ideals of . Note
that the f {FC property coincides with the ~{FC property, because M( f ) = M(~)
(Proposition1.4(5)).
Proof.
A
T
N
?
?; ?
ATN
N
TN
? f
N
D
ADN
?
D
D
N
D
D
ADN
CT
D DN
?0
A
A
CT
T
A
D
?0
A
D
?
N
D
D
?; ?
D
D
DN
C
?
A
?
C DN
D
D
?
C DN
D
CT
D
T
?
?
D
?
x
A
?
C
D
D
D
D
A
D
C T TN
D
?
C
ATN
? f
?0
DN
T
?; ?
A
N
D
?
? f
?
D
?
?
D
?
?
?
Let D be an integral domain and ? a semistar operation on D.
is ~?{Noetherian, then DM is Noetherian, for each M 2 M(?f ). Moreover, if
has the ?f {FC property, then the converse holds.
Proposition 3.8.
If
D
D
Let 2 M( f ), an ideal of and := \ . Since is ~{Noetherian,
there exists a nitely
generated
ideal of with ~ = ~ (Lemma 3.3). Then,
~
~
=
=
=
=
(we used twice the fact that ~ is spectral,
dened by M( f )). Then is a nitely generated ideal of
and so
is
Noetherian. For the converse, assume that the f {FC property holds on . Let
be a nonzero ideal of and let 0 6= 2 . Let 1 2
2 M( f )
be the quasi{ f {maximal ideals containing . Since i is Noetherian for each
=1 2
, then i =
of .
i , for some nitely generated ideal
Proof.
M
?
A
DM
J
A
I DM
?
I DM
J
?
DM
?
I
I
A
D
D
J
?
D
I
?
?
J DM
?
A
DM
DM
?
I
D
x
?
i
;
;::: ;n
I
x
I DM
Ji DM
D
M ; M ; : : : ; Mn
?
DM
Ji
I
D
16
The ideal := + 1 + 2 + + of is nitely generated and contained in . It
is clear that, for each = 1 2
, i=
2 M( f ) and
i . Moreover, if
6= , for each T= 1 2
,
then
62 Tand this fact implies
=
=
. Then, ~ = f
j 2 M( f )g = f
j 2 M( f )g = ~ . Thus,
by Lemma 3.3, is ~{Noetherian.
Remark 3.9. (1) Note that Proposition 3.8, in case of star operations, can be
deduced from [25, Proposition 4.6], proven in the context of weak ideal systems on
commutative monoids.
(2) Note that strong Mori domains (that is, {Noetherian domains, where :=
~) or, more generally, Mori domains satisfy always the {FC property (= {FC
property, since M( ) = M( ), for every integral domain) by [6, Proposition 2.2(b)].
But it is not true in general that the ~{Noetherian domains satisfy the f {FC
property (take, for instance, := Z[ ], := , and observe that is contained in
innitely many maximal ideals of Z[ ]).
Note that, from Proposition 3.8 and from the previous considerations, we obtain in
particular that an integral domain is strong Mori if and only if is Noetherian,
for each
2 M( ), and has the {FC property (cf. also [45, Theorem 1.9]).
4. Semistar Dedekind domains
Let be an integral domain and a semistar operation on . We recall from
Section 1 (or [16, Section 2]) that a nonzero fractional ideal (2 F ( )) of
is {invertible if ( 1) = and 2 F ( ) is quasi{ {invertible if ( ( :
)) = (note that, the last property implies that 2 F ( )). It is clear that
a {invertible ideal is quasi{ {invertible. The converse is not true in general [16,
Example 2.9 and Proposition 2.16] but, if is stable (e.g., for = ~), a nitely
generated ideal is {invertible if and only if it is quasi{ {invertible [16, Corollary
2.17(2)].
B
xD
J
J
i
M
Mi
i
DM
I
;
?
:::
;
D
I
I DM
;::: ;n
I DM
D
Jn
;::: ;n
x
BDM
M
M
M
?
I DM
?
BDM
M
?
BDM
B
?
?
w
w
v
t
w
w
t
?
D
X
?
?
d
X
X
D
M
t
D
DM
w
D
?
D
F
?
?
E
?
FF
D
D
?
E
D
?
E
?
D
D
?
E D
D
?
?
?
?
?
?
?
?
Let D be an integral domain and ? a semistar operation on D.
The following are equivalent:
(1) D is a ?{Noetherian domain and a P?MD;
(1f ) D is a ?f {Noetherian domain and a P?f MD;
(2) F ~?(D~) := f~ F ?~ j F~ 2~F~(D)g is a~ group under the multiplication \", dened
by F ? G? := (F ? G? )? = (F G)?, for all F; G 2 F (D);
(3) Each nonzero fractional ideal of D is quasi{~?{invertible;
(4) Each nonzero (integral) ideal of D is quasi{~?{invertible.
Proposition 4.1.
(1) , (1f ) is obvious (Proposition 3.5 and Proposition 1.6 ((i),(vi))).
(1) ) (2). One can easily check that F ~( ) is a monoid, with ~ as the identity
element (with respect to \"). We next show that each element of F ~( ) is
invertible for the monoid structure. Let 2 F ( ), then there exists 0 6= 2
such that := . Write f = f , where is a nitely generated
ideal of (Lemma 3.3 and Proposition 3.5). Since is a P MD, then f = ~
(Proposition 1.6). So, ~ = ~. We have ( 1)~ = ~, since is a P~MD
1)~ = ~( 1)~ ~ .
(Proposition 1.6). Then, ~ = ( ~ 1)~ = ( 1)~ = (
Thus ~ is invertible in (F ~( ) ).
(2) )~ (3). ~ Let 2 F ( ). By~ assumption,~ there exists ~ 2 F ( ~) such that
( ) =~ . We have ~ ~ , so~ ( : ). Thus = ( ) ( ( ~ :
)) . Hence ( ( : )) = , that is, is quasi{~{invertible.
(3) ) (4) is straightforward.
(4) ) (1) From the previous comments on quasi semistar invertibility for nonzero
nitely generated ideals in the stable case, it is clear that the assumption implies
Proof.
?
D
D
?
?
F
I
dF
D
I
?
J
D
?
J
D
I
J
D
F
F
FG
F
?
D
D
?
?
?
?
?
?
JJ
?
?
J J
?
IJ
?
D
F D
?
D
?
?
?
F
?
dJ
?
?
D ;
G
FG
D
?
?
F
?
dF J
D
?
D
I
D
?
D
d
?
G
D
?
D
?
F
F
D
?
?
D
FG
?
F D
?
SEMISTAR DEDEKIND DOMAINS
17
that is a P~MD and hence is a P MD (Proposition 1.6). To prove that is a
{Noetherian domain, since ~ = f (Proposition 1.6), it is enough to show, by using
Proposition 3.5, that
is ~{Noetherian. Let be a nonzero ideal of , then, by
assumption, ( ( ~ : ))~ = ~ . By [16, Lemma 2.3 and Proposition 2.15] applied
to~ ~, there
exists a nonzero nitely generated ideal of such that and
= ~ . From Lemma 3.3, we deduce that is ~{Noetherian.
An integral domain with a semistar operation satisfying the equivalent
conditions (1){(4) of Proposition 4.1 is called a {Dedekind domain ( {DD, for
short). Note that, by denition, the notions of {DD and f {DD coincide.
Remark 4.2. (1) By Proposition 4.1(1), if = we obtain that a {DD coincides
with a classical Dedekind domain [19, Theorem 37.1]; if = , we have that a
{DD coincides with a Krull domain (since a Mori P MD is a Krull domain [33,
Theorem 3.2 ((1) ,(3))]; note that a Mori domain veries the {FC property by
[6, Proposition 2.2(b)]). More generally, if is a star operation, then is a {DD
if and only if is {Dedekind in the sense of [23, Chapter 23].
(2) If is {DD then is {ADD (for a converse, see the following Theorem
4.11). Indeed, since a {DD is a P MD and so ~ = f (Proposition 1.6). This equality implies also that is ~{Noetherian (Proposition 3.5 and Proposition 4.1(1)).
Therefore is Noetherian (by Proposition 3.8) and, hence, we conclude that
is a DVR, for each 2 M( f ).
D
?
D
?
?
D
I D
?
D
?
?
I
?
?
D
I
D
?
?
J
J
?
I
?
D
D
J
I
?
D
?
?
?
?
?
?
d
d
?
v
v
v
t
?
D
D
D
?
?
?
D
?
?
?
D
?
?
?
DM
DM
M
?
Let D be an integral domain and
is a ?{DD if and only if D is a ~?{DD.
Corollary 4.3.
Then
D
?
a semistar operation on
D
.
It follows fromProposition 4.1(4) and from the fact that ~ = ~, since M(e) =
M( f ) (cf. also [12, page 182]).
Proof.
?
?
?
?
Theorem 4.4.
(1)
Let
D
be an integral domain.
Let ? ?0 be two semistar operations on D. Then:
0
D is a ?{DD ) D is a ? {DD .
In particular:
(1a) If is a Dedekind domain, then is a {DD, for any semistar operation
on .
(1b) Assume that is a (semi)star operation on . Then a {DD is a Krull
D
?
D
?
D
?
(2)
D
?
domain.
Let T be an overring of D. Let ? be a semistar operation on D and ?0 a
semistar operation on T . Assume that T is a (?; ?0){linked overring of D. If
0
?
D is a ?{DD, then T is a ? {DD. In particular, If D is a ?{DD, then D
is
a ?_ {DD.
(1) follows from [11, p. 30] and Lemma3.1(1). (1a) and (1b) are consequence
of (1), Remark 4.2(1) and of the fact that , for each semistar operation , and
if is a (semi)star operation, then .
(2) Note that if is a ( 0){linked overring of and if is a P MD, then is a
( 0){at over [9, Theorem 5.7 ((i))(ii))]. By Proposition 4.1(1) and Corollary
4.3, we know that is ~{Noetherian and a P MD (or, equivalently, a P~MD).
Hence, is e0{Noetherian (Proposition 3.7) and is a P 0MD (or, equivalently, a
P ~0MD) by [9, Corollary 5.4]. The rst statement follows from Proposition 4.1(1)
and Corollary 4.3. The last statement is a consequence of [9, Lemma 3.1(e)].
Proof.
d
?
?
T
?; ?
?; ?
?
?
v
D
D
?
T
D
D
T
?
?
?
?
T
?
?
Let D be an integral domain and
. Then the following are equivalent:
(1) D is a ?{DD
(2) D is a Krull domain and ?f = t
Proposition 4.5.
D
?
a (semi)star operation on
18
(1) ) (2). By Theorem 4.4(1b), if is a {DD, then is a Krull domain,
in this case, f = [11, Proposition 3.4].
(2) ) (1). This follows from Remark 4.2(1) and from the fact that {DD = {DD
= f {DD = {DD.
Note that Proposition 4.5 has already been proven in [23, Theorem 23.3((a),(d))],
by using the language of monoids and ideal systems.
Remark 4.6. Note that if
is {DD, then by Theorem 4.4(2) is _ {DD, that
is is a Krull domain and (_ ) = ? (Proposition 4.5). However, the converse
does not hold in general as the example in Remark 2.3(2) shows. Nevertheless, the
converse is true when = ~ (see the following Corollary 4.20) or when the extension
is at, as a consequence of Lemma 3.1(2) and [11, Proposition 3.2]. For a
more accurate discussion on this problem see the following Remark 4.21.
Next result is a \Cohen-type" Theorem for quasi{ {invertible ideals.
Proof.
D
?
?
D
t
v
?
D
D
?
?
D
? f
?
D
t
?
D
?
?
tD
?
?
?
Let D be an integral domain and ? a semistar operation of nite type
. The following are equivalent:
(1) Each nonzero quasi{?{prime of D is a quasi{?{invertible ideal of D.
(2) Each nonzero quasi{?{ideal of D is a quasi{?{invertible ideal of D.
(3) Each nonzero ideal of D is a quasi{?{invertible ideal of D.
Lemma 4.7.
on
D
(1) ) (2). Let be the set of the quasi{ {ideals of that are not quasi{
{invertible. Assume that 6= ;. Since = f by assumption, then Zorn's Lemma
can be applied, thus we deduce that has maximal elements. We next show that a
maximal element of is prime. Let be a maximal element of and let 2 ,
with 2 . Suppose 62 . Let := ( : ). We claim that \ = .
Indeed, since ( : ) ( : ? ), then \ ( : ? ) \ =
( : ). Moreover, if 2 ( : ), then 2 \ = , and hence
( : ) ( : ) = . Thus = \ , i.e. is a quasi{ {ideal of
. Clearly, contains properly (since 2 r ). By the maximality of in
, it follows that is quasi{ {invertible, that is ( ( : )) = . We notice
that ( : ) 2 F ( ) is not quasi{ {invertible, since is not quasi{ {invertible
[16, Lemma 2.10]. We deduce that ( ( : )) \ is a proper quasi{ {ideal,
that is not quasi{ {invertible [16, Remark 2.13(a)] and, obviously, it contains .
From the maximality of in , we have ( ( : )) \ = . Now, implies ( ) . Then 2 ( ) = ( ( : )) ( ( : )) . Therefore,
2 ( ( : )) \ = and so we have proven that is a prime ideal of .
(2) ) (3) is a consequence of [16, Remark 2.13(a)], after remarking that, for each
nonzero ideal of , then := \ , where is a quasi{ {ideal of and
= .
(3) ) (2) ) (1) are trivial.
Remark 4.8. Note that, in the situation of Lemma 4.7, the statement:
(0) each nonzero quasi{ {maximal ideal of is a quasi{ {invertible ideal of ,
is, in general, strictly weaker than (1). Take, for instance, equal to a discrete
valuation domain of rank 2, and = .
The next two theorems generalize [19, Theorem 37.8 ((1),(4)), Theorem 37.2].
Similar results are proven in [23, Theorem 23.3((a),(c), (h))].
Proof.
S
?
?
S
?
D
?
S
S
rs
P
s
P
P
P
?
?
D
D
P
rD
D
P
?
rD
D
P
x
rD
P
D
rD
D
?
P
J
J
?
D
xr
J
J
s
?
J
P
P
D
?
?
?
P
?
?
?
J
D
?
P
?
J
D
P
J
J D
P D
D
D
rD
D
D
?
P
?
D
J
J
J
rD
D
r; s
D rD
rD
P
S
?
?
P
J
P D
S
J
?
J
?
D
?
?
P
P
rJ
r
P D
?
?
P
?
J
J
J
?
I
?
D
S
r
P D
rD
?
rJ D
?
?
J
J
?
P
D
?
D
P D
P
?
rJ
J
P
J
I
J
?
D
P
?
D
I
?
D
?
?
D
?
D
D
?
dD
Let D be an integral domain and ? a semistar operation on D. The
following are equivalent:
(1) D is a ?{DD.
(2) Each nonzero quasi{~?{prime ideal of D is quasi{~?{invertible.
Theorem 4.9.
SEMISTAR DEDEKIND DOMAINS
19
Easy consequence of Lemma 4.7 ((1),(3)) and Proposition 4.1 (4).
From the previous theorem, we deduce the following characterization of Krull
domains (cf. [31, Theorem 2.3 ((1),(3))], [33, Theorem 3.6 ((1),(4))] and [44,
Theorem 5.4 ((i),(vi))]).
Proof.
Let D be an integral domain. The following are equivalent:
D is a Krull domain.
Each nonzero w{prime ideal of D is w{invertible.
Each nonzero t{prime ideal of D is t{invertible.
Corollary 4.10.
(1)
(2)
(3)
(1) , (2) is a direct consequence of Theorem 4.9.
(1) ) (3) is a straightforward consequence of (1) ) (2) and of the fact that, in a
Krull domain (which is a particular P MD), = e = (Proposition 1.6).
(3) ) (2). Note that, by assumption, and by Lemma 4.7 ((1),(3)),1 every nonzero
ideal of is1 {invertible. Let be a nonzero {prime. If (
) 6= , then
(
)
, for some 2 M( ) = M( ) (Proposition 1.4(5)), thus
( 1) = (( 1) ) = , which is a contradiction.
Proof.
v
D
Q
t
t
QQ
Q
w
QQ
t
M
M
t
w
w
M
w t
QQ
t
w
w
QQ
D
t
M
Let D be an integral domain and ? a semistar operation on D.
The following are equivalent:
(1) D is a ?{DD.
(2) D is a ?{ADD and each nonzero element of D is contained in only nitely
many quasi{?f {maximal ideals (i.e. D has the ?f {FC property).
(3) D is a ?{Noetherian ?{ADD.
Theorem 4.11.
(1) ) (2). Clearly is a {ADD, by Remark 4.2(2). Since each quasi{
{maximal ideal of is a contraction of a _ {maximal ideal of [15, Lemma
2.3(3)], in order to show that has f {FC property, it is enough to check that
satises the _f {FC property. On the other hand, since (1) implies that is a _ -DD
(Theorem 4.4(2)), without loss of generality, we can assume that is a (semi)star
operation on and is a {DD. By Proposition 4.5, is a Krull domain and
f = . Thus, each nonzero element is contained in only nitely many {maximal
ideals (= f {maximal ideals) of .
(2) ) (1). We need to show that is f {DD. First, note that is a P f MD
and is Noetherian, for each 2 M( f ) (Proposition 2.2 (1) and (2)). The
conclusion now follows from Proposition 3.8 and Proposition 4.1(1), after recalling
that, in a P f MD, f = e (Proposition 1.6).
(1) , (3) is a consequence of Proposition 2.2(2), Proposition 4.1 and Remark
4.2(2).
From the previous theorem, we deduce a restatement of a wellknown characterization of Krull domains:
Proof.
D
?f
?
D
?f
D
D
?
?
D
?
D
?
?
?
?
D
?
D
?
D
t
t
?
D
D
DM
M
?
?
?
D
?
?
?
Let D be an integral domain, then the following are equivalent:
D is a Krull domain.
D is a t{almost Dedekind domain and each nonzero element of D is contained
in only nitely many t{maximal ideals (= t{FC property).
2
Corollary 4.12.
(1)
(2)
Let be an integral domain and a semistar operation on . We recall that
the {integral closure [ ] of (or, the semistar integral closure with respect to
the semistar operation of ) is the integrally closed overring of
dened by
[ ] := f( : ) j 2 f ( )g [13, Denition 4.1]. We say that is quasi{
{integrally closed (respectively, {integrally closed ) if
= [ ] (respectively,
[
]
= ). It is clear that:
D
?
?
D
?
D
?
F
?
D
?
F
?
F
?
D
D
D
?
?
D
?
D
D
D
D
?
D
?
20
{ is quasi{ {integrally closed if and only if is quasi{ f {integrally closed
(respectively, is {integrally closed if and only if is f {integrally closed);
{ is {integrally closed if and only if is quasi{ {integrally closed and is a
(semi)star operation on .
Note that when = , then the overring [ ] = [ ] was studied in [3] under
the name of psedo-integral closure of .
D
?
D
D
D
?
?
D
?
D
?
?
?
D
?
v
v
D
D
t
D
Let D be an integral domain and ? a semistar operation on
is e.a.b., then D? = D[?] (i.e. D is quasi{?{integrally closed).
is quasi{~?{integrally closed if and only if D~? is integrally closed.
Lemma 4.13.
(1)
(2)
If
D
D
.
?
(1) Note that, in general, [ ] . For the converse, let 2 f ( ) and
let 2 ( : ). Then,
and = + ( ). Therefore we have
( ( + )) = ( ( + )) = ( + ( )) = . From the fact that is
nitely generated and that is e.a.b., we obtain ( + ) = . It follows that
2 and so ( : ) . Hence, = [ ] .
(2) The \only if" part is clear. For the \if" part, let 0 be the integral closure of
, since ~ is integrally closed, then ( 0 )~ ~ [~] hence, by [11, Example
2.1(c2)], ( 0 )~ = ~ = [~] . Therefore, is quasi{~{integrally closed.
Proof.
x
D
F
F D
?
F
?
xF
?
xD
F
?
D
?
xD
F
?
?
F
?
D
?
F
?
F
F
?
?
F
D
?
F
?
F
?
D
F
?
xD
?
x
?
D
?
D
D
xD
?
F
D
?
?
xD
F
?
D
?
?
D
D
D
?
D
?
D
D
?
D
?
?
D
D
?
D
?
?
Let ? be a semistar operation on an integral domain
a P?MD (in particular, a ?{DD) then D is quasi{?{integrally closed.
Corollary 4.14.
D
. If
is
D
It follows from Lemma 4.13(1) and from the fact that, in a P MD, ~ = f
is an e.a.b. semistar operation (Proposition 1.6 ((i))(v), (vi))).
The following result shows that a semistar version of the \Noether's Axioms"
provides a characterization of the semistar Dedekind domains.
Proof.
?
?
?
Let D be an integral domain and ? a semistar operation on
The following are equivalent:
(1) D is a ?{DD.
(2) D is ~?{Noetherian, ~?- dim(D) = 1 and D is quasi{~?{integrally closed.
(3) D is ~?{Noetherian, ~?- dim(D) = 1 and D~? is integrally closed.
Theorem 4.15.
D
.
The equivalence (2) , (3) follows from Lemma 4.13 (2).
(1) ) (2). Since is a {DD, then is {ADD (Remark 4.2(2)). Hence ~dim( ) = 1 (Proposition 2.16). Moreover, recall that a {DD is a ~{DD (Corollary
4.3). Then is ~{Noetherian and a P~MD (Proposition 4.1), and so is quasi{
~{integrally closed by Corollary 4.14.
(3) ) (1) For each 2 M( f ), it is wellknown that ~ and ~ M \ ~? =
. Since ~ is integrally closed, this implies that is also integrally closed.
Therefore
is a local, Noetherian (by Proposition 3.8), integrally closed, one
dimensional (by Lemma 2.11) domain, that is, a DVR [19, Theorem 37.8]. Hence
is a P MD. In particular, we have ~ = f (Proposition 1.6), thus is f {
Noetherian, by the assumption, and so is {Noetherian (Proposition 3.5). We
conclude that is a {DD.
By taking = in Theorem 4.15, we obtain the following characterization of
Krull domains:
Proof.
D
?
D
?
?
D
?
D
?
?
?
D
?
M
DM
D
?
D
?
?
DM
?
D MD
D
DM
DM
D
?
D
?
D
D
D
D
is
is
is
is
a
a
a
a
?
D
?
D
?
v
Let D be an integral domain. The following are equivalent:
Krull domain.
strong Mori domain, w- dim(D) = 1 and D = D[w] .
strong Mori domain, w- dim(D) = 1 and D is integrally closed.
strong Mori domain, t- dim(D) = 1 and D is integrally closed.
Corollary 4.16.
(1)
(2)
(3)
(4)
?
?
SEMISTAR DEDEKIND DOMAINS
21
The only part which needs a justication is the statement on -dimension
and -dimension (in the equivalence (3) , (4)). This follows from the fact that,
in every integral domain, and M( ) = M( ).
Remark 4.17. Note that, if
is a {DD, then we know that ~ = f , and so
satises the properties:
(2f ) is f {Noetherian, f - dim( ) = 1 and is quasi{ f {integrally closed;
(3f ) is f -Noetherian, f - dim( ) = 1 and f (= ) is integrally closed
obtained from (2) and (3) of Theorem 4.15, replacing ~ with f . But, conversely,
if satises either (2f ) or (3f ) then is not necessarily a {DD. Indeed, let
and be as in the example of Remark 2.3(2). Then we have already observed that
= f and ~ = . Moreover, is not a {DD (because it is not a {ADD), but
f =
= [ f ] is integrally closed (since is a DVR), f -dim( ) = 1 (since
M( f ) = f g and f -dim( ) dim( ) = 1) and is f {Noetherian (Lemma
3.3, since is Noetherian).
Note that (3f ) does not imply that is a {DD, even if is a (semi)star operation
on . Take and as in the example described in Remark 2.3(2) and, moreover,
assume that is algebraically closed in . It is wellknown that, in this situation,
is integrally closed. Let := on . It is easy to see that M( ) = M( ) = f g,
thus = is the identity (semi)star operation on (hence, [ ] = [ ] = )
and - dim( ) = 1 (= - dim( ) = -dim( ) = dim( )). Moreover, it is known
that is a Mori domain [17, Theorem 4.18] and thus is a {Noetherian domain.
However, is not a Krull domain, since is not completely integrally closed (being
the complete integral closure of ). Note that, in this situation, is even not a
strong Mori domain (by Corollary 4.16).
Note also that, in the previous example, ( [ ] (i.e. is not {integrally closed,
hence does not satises condition (2f ) for = ), since [ ] = by [3, Theorem
1.8(ii)].
On the other hand, if is a (semi)star operation on , then we know that is
a {DD if and only if is a {DD (i.e. a Krull domain) and f = (Proposition
4.5). It is interesting to observe that, for = , condition (1) of Theorem 4.15 is
equivalent to (2f ). More precisely we have the following variation of the equivalence
(1) , (4) of Corollary 4.16:
is a Krull domain if and only if
is {Noetherian, - dim( ) = 1 and
is
{integrally closed (i.e. = [ ] ).
As a matter of fact,
let1 2 f ( ),1 then
= [ ] 1= [ ] implies that =
1
1
( : )=( : )=(
) and so (
) = . Moreover, since
{Noetherian is equivalent to {Noetherian (Proposition 3.5) and {Noetherian
implies that = (Lemma 3.3), then ( 1) = . Thus is a P MD and so
is a {DD (Proposition 4.1).
Finally, from the previous considerations we deduce that is a {DD if and
only if
(2f ) is f {Noetherian, f - dim( ) = 1, is quasi{ f {integrally closed and
=
.
f
We conclude with a question: is there an example of an integral (Krull) domain
, equipped with a (semi)star operation , such that condition (2f ) holds but (2f )
does not? Note that if such an example exists then necessarily f ( ) [19,
Theorem 37.8 ((1),(2))].
Next result generalizes [19, Proposition 38.7].
Proof.
t
w
w
t
t
D
w
?
?
D
?
?
D
D
D
?
?
D
D
D
?
?
D
?
?
D
?
?
D
?
D; T
?
?
D
?
?
?
T
dD
D
?
D
?
?
?
T
M
?
D
?
D
D
D
?
T
D
D
T
k
?
K
?
w
?
D
v
D
D
v
d
t
D
D
v
D
w
D
D
w
D
D
D
D
?
t
D
v
?
D
D
D
D
v
t
D
F
t
D
FF
t
t
D
D
t
F
F
D
v
?
v
T
D
D
t
t
t
D
?
?
F
D
t
D
F
d
D
T
v
D
M
D
D
D
t
D
t
FF
D
v
v
D
D
v
v
v
t
FF
t
D
D
v
D
v
D
D
?
?
?
D
D
?
?
t
D
?
d
Let D be an integral domain and
The following are equivalent:
Theorem 4.18.
?
?
t
a semistar operation on
D
.
22
(1) is a {DD.
(2) Na( ) (= Kr( )) is a PID.
(3) Na( ) (= Kr( )) is a Dedekind domain.
Proof. (1) ) (2). Since is a P MD, then Na(
) = Kr( ) is a Bezout domain (Proposition 1.5 ((i))(iv)) and Proposition 1.4(1)). Now, let I be a nonzero
ideal of Na( ) and set := I \ . We claim that I = Na( ). The inclusion
Na( ) I is clear. For the opposite inclusion, since I = (I \ [ ]) Na( ),
it is enough to show that I \ [ ] Na( ). Let 2 I \ [ ], then
Na( ) = Kr( ) = c( )Kr( ) = c( )Na( ) (where the second equality
holds by Proposition 1.5(6)). Hence c( ) Na( ) \ I \ = . Therefore we conclude that 2 c( )Na( ) Na( ), which proves our claim.
Now, since is a ~{Noetherian domain (as is a {DD, cf. Corollary 4.3 and
Proposition 4.1), then ~ = ~ for some 2 f ( ), with (Lemma 3.3).
Since ~ = Na( ~ ) \ , for each
2 ( )(Proposition 1.4(4)), then we have
I = Na( ) = Na( ) = ~Na( ) = Na( ). Hence we conclude that
I is a principal ideal in Na( ), because, as we have already remarked, Na( )
is a Bezout domain.
(2) ) (3) is trivial.
(3) ) (1). Assume that Na( ) is a Dedekind domain then, obviously, Na( ) =
Kr( ) (Proposition 1.6 ((i) ) (iv))) and Na( ) is an ADD, and hence is a
{ADD (Theorem 2.5). In order to apply Theorem 4.11, it remains to show that
has the f {FC property. Let 0 6= 2 . Since Max(Na( )) = f Na( ) j
2 M( f )g (Proposition 1.4(2)) and Na( ) is a Dedekind domain, then there
are only nitely many maximal ideals Na( ) containing . Furthermore,
Na( ) \ = , for each 2 M( f ) = M(e) (Proposition 1.4(4)). Hence
is contained in only nitely many quasi{ f {maximal ideals of . Therefore we
conclude that is a {DD.
From the previous result, we deduce immediately:
D
?
D; ?
D; ?
D; ?
D; ?
D
D; ?
I
?
I
D; ?
D; ?
D
I
D; ?
D; ?
D X
D X
f
D; ?
f
D; ?
f
I
D; ?
D; ?
f
f
f
D
?
I
D; ?
E
D; ?
D; ?
I
D; ?
I
?
F
?
F
K
E
D; ?
F
?
D
D
I
D; ?
D
?
D; ?
D X
D; ?
f
?
I
E
f
f
?
D
F
I
F D
D; ?
F
D; ?
D; ?
D; ?
D; ?
D; ?
D; ?
D; ?
D
?
D
?
M
?
M
D; ?
x
D
D; ?
M
D; ?
D; ?
M
D
M
M
D; ?
?
x
x
?
?
D
D
?
Let D be an integral domain. The following are equivalent:
D is a Krull domain.
Na(D; v) (= Kr(D; v)) is a PID.
Na(D; v) (= Kr(D; v)) is a Dedekind domain.
Corollary 4.19.
(1)
(2)
(3)
Another consequence of Theorem 4.18 is the following:
Let D be an integral domain and
The following are equivalent:
(1) D is a ?{DD.
(2) D is a ~?{DD.
(3) D~? is a ~?_ -DD.
(4) D~? is a Krull domain and ~?_ = tD~? .
Corollary 4.20.
?
a semistar operation on
2
D
.
The equivalence (1) , (2) , (3) follows from Theorem 4.18 and the fact
that Na( ) = Na( ~) = Na( ~ ~_ ) (Proposition 1.4(6)).
The equivalence (3) , (4) follows from Proposition 4.5, using the fact that (1) ,
(3).
Remark 4.21. From Corollary 4.20 ((1),(4)), we have that if
is a {DD then
:= ~ is a Krull domain and ~ = ( . )D (where is the {operation of ). Note
that it is not true in general that, if is a Krull overring of an integral domain ,
then is a ( . )D {Dedekind domain .
For instance, let be a eld
and an indeterminate over . Set := [ ] ,
:= and := [ 2 3] . It is easy to see that is a one-dimensional
Proof.
D; ?
?
D; ?
D ;?
D
T
D
?
?
tT
tT
?
t
T
T
D
K
M
XT
D
tT
D
X
K X ;X
K
D
T
K X
SEMISTAR DEDEKIND DOMAINS
23
local Noetherian integral domain with integral closure equal to and maximal
ideal equal to := \ (with = ). Therefore, in this case, =
is the identity (semi)star operation on and so the semistar operation ( . )D on
coincides with f g. Clearly ]
( f g), since M( f g) = f g and,
f g=
obviously = is not a DVR. Therefore is not a ( . )D {Dedekind domain.
From the positive side, we have the following answer to the question of when,
given a Krull overring of an integral domain , is a ( . )D {DD:
T
N
M
D
NT
N
tT
T
D
? T
D
? T
dD
DN
? T
Let
T
? T
D
T
(4.21.1)
N
tT
D
D
be an overring of an integral domain
equivalent:
(1) D is a (t.T )D {DD.
(2) T is a Krull domain and, for each
dT
tT
tT
D
. The following are
= .
The previous characterization is a straightforward consequence of the following
\restatement" of the equivalence given in Corollary 4.20 ((1),(4)):
tT
{maximal ideal
Q
of T ,
\
DQ D
TQ
(4.21.2) If D is an integral domain and ? is a semistar operation on
the following are equivalent:
(1) D is a ?{DD.
(2) There exists an overring T of D such that T is a Krull domain, ?f
and, for each tT {maximal ideal Q of T , DQ\D = TQ .
D
, then
= ( . )D
tT
To show the previous equivalence, note that in general the set of the quasi{( . )D {
maximal ideals in coincide with the set f \ j is a {maximal ideal in g
[15, Lemma 2.3(3)]. Therefore the assumption that f = ( . )D and, for each {
maximal ideal of , \ = implies that ~ = ( ) T , for each 2 F ( ),
(in particular, ~ = ), and so ~_ = . Therefore (4.21.2(2)) implies condition
(4) of Corollary 4.20.
Conversely, assume that condition (4) of Corollary 4.20 holds and set := ~ .
Since ~_ = and is a {DD (Corollary 4.20 ((4))(1))), then ~ = f (Proposition
1.6 and 4.1), and so _f = . Therefore f = ( . )D (Proposition 1.3(2)). Moreover,
by the previous considerations, the set of the quasi{ f {maximal ideals in coincide
with the set f \ j is a {maximal ideal in g. Since is a {DD and hence,
in particular, a {ADD (and since is a Krull domain), then \ is a DVR, which
must coincide with its (DVR) overring , for each {maximal ideal of .
It is possible to give another proof of (4.21.2) by using Lemma 3.1(2)) and showing the following preliminary result of intrinsical interest concerning the P MDs:
tT
D
Q
D
Q
tT
?
Q
D
T
?
DQ D
TQ
T
?
T
E
?
T
tT
ET
tT
t
E
D
tT
T
?
tT
D
?
?
?
tT
?
D
Q
tT
?
D
T
?
?
tT
?
Q
D
D
T
?
DQ D
TQ
tT
Q
T
?
Let D be an integral domain and ? a semistar operation on D. Then,
the following are equivalent:
(1) D is a P?MD.
(2) There exists an overring T of D such that T is a PvT MD, ?f = (t.T )D and,
for each tT {maximal ideal Q of T , DQ\D = TQ .
(4.21.3)
The proof is based on a variation of the techniques already discussed above and
the details are omitted.
Example 4.22. Let
be a Mori domain, let be the set of all the maximal {
ideals of which are {invertible and let be the spectral semistar operation on
associated to (Example 1.1 (3)). Assume that 6= ; (i.e. that is a Mori
non strongly Mori domain, accordingly to the terminology introduced by Barucci
and Gabelli [6, page 105]), then is a {DD.
We apply the characterization given in (4.21.1) or in Corollary 4.20 ((1),(4)).
Note that by [6, Proposition 3.1 and Theorem 3.3 (a)], is a Krull domain such
that the map 7! denes a bijection between and the set M( ? ) of
all the {maximal ideals of and = ( ) ? . Therefore the (semi)star
D
D
t
t
?
D
D
D
?
D
P
t
P
?
?
tD
D
?
DP
D
?
P
24
operation _ on coincides with the {operation, ? , on . Moreover, it
is easy to see that, on , the semistar operation (t. ? )D coincides with .
?
D
?
t
tD
D
D
?
?
D
Let be an integral domain and a semistar operation on . We say that two
nonzero ideals and are {comaximal if ( + ) = . Note that, if is a
semistar operation of nite type, then and are {comaximal if and only if
and are not contained in a common quasi{ {maximal ideal.
D
?
A
B
D
?
A
A
?
B
B
B
?
D
?
?
A
?
Let D be an integral domain and ? a semistar operation on D. Let
be two nonzero ?{comaximal ideals of D. Then (A \ B )? = (AB )? .
Lemma 4.23.
A
and
B
In general ( + )( \ ) . Then, (( + )( \ )) ( ) ( \ ) . But (( + )( \ )) = (( + ) ( \ )) = ( ( \ )) = ( \ ) .
Hence, ( \ ) = ( ) .
Proof.
A
A
?
B
A
A
B
?
B
B
A
Let
B
AB
?
B
A
A
?
B
A
B
?
B
A
D
?
?
B
A
B
?
AB
?
A
?
B
?
AB
Corollary 4.24.
A
be an integral domain and
D
a semistar operation of nite
?
2 and let 1 2
be nonzero ideals of , such that ( +
) = , for =6 . Then, ( 1 \ 2 \ \ ) = ( 1 2 ) .
Proof. We prove it by induction on 2, using Lemma 4.23 for the case = 2.
Set := 1 \ 2 \ \ 1 and := . Then, and are not contained in
a common quasi{ {maximal ideal, otherwise, and (for some 1 1)
would be contained in a common quasi{ {maximal ideal. Hence ( 1 \ 2 \ \
) =( 1 2 ) .
1\ ) =( \ ) =( ) =(
type. Let
?
Aj
D
n
A ; A ; : : : ; An
?
i
j
A
A
D
:::
?
An
A A
Ai
:::
An
?
n
A
A
A
:::
An
n
B
An
A
?
An
B
Aj
j
?
An
?
An
A
B
?
?
AB
A
?
?
A B
A A
:::
n
A
:::
?
An
Let D be an integral domain and ? semistar operation. The following are equivalent:
(1) D is a ?-DD.
(2) For each nonzero ideal I of D, there exists a nite family of quasi{?f {prime
ideals P1 ; P2; : : : ; Pn of D, pairwise ?f -comaximals, and a nite family of non
negative integers e1 ; e2 ; : : : ; en such that I ~? = (P1e1 P2e2 : : : Pnen )~? .
Moreover, if (2) holds and if I ~? 6= D~? , then we can assume that Pi ~? 6= D~? , for
each i = 1; 2; : : : ; n. In this case, the integers e1 ; e2; : : : ; en are positive and the
factorization is unique.
Theorem 4.25.
(1) ) (2). Let be a nonzero ideal of . To avoid the trivial case, we
can assume that ~ 6= ~. Let 1 2
be the nite (non empty) set of
quasi{ f {maximal ideals such that , for 1 (Theorem 4.11). We
have ~ = \f j 2 M( f )g = \ ==1 ( i \ ~ ). Since i is a DVR, then
i
1, = 1 2
. Therefore, we have
i =
i , for some integers
~
~
i
i )~ . Hence ~ = (
1 )~ \ ( 2 )~ \ \ ( n )~ =
\
=
\
=
(
i
i
1
2
( 1 1 \ 2 2 \ \ n )~ = ( 1 1 2 2 n )~, by Corollary
4.24.
For the last statement, let ~ = ( 1 1 2 2 n )~, if~ ~ = ~ , for some , then
obviously we can cancel from the factorization of . ~
We prove next the uniqueness of the representation
of . From (Proposition
1.4(4)), we deduce that Na( ) = 1 1 2 2 n Na( ) = ( 1Na( )) 1
( 2Na( )) 2 ( Na( )) n is the unique factorization into primes of the
ideal Na( ) in the PID Na( ) (Theorem 4.18). Since = Na(~ ) \
(because each is a quasi{e{maximal ideal of ), the factorization of is unique.
(2) ) (1) Without loss of generality, we can assume that is not a eld. First,
we prove that each localization to a quasi{ f {maximal ideal of is a DVR. Let
2 M( f ) and let be a nonzero proper ideal of . Set := \ ( ).
Then, it is easy to see that ~ 6= ~ thus, by assumption, ~ = ( 1 1 2 2 n )~,
for some family of quasi{ f {prime ideals , with ~ 6= ~ and for some family
Proof.
I
I
?
D
D
?
P ; P ; : : : ; Pn
?
I
I
?
e
I DP
P
D
e
P
?
e
P
e
i
:::
Pi
i
i
?
Pi DP
I DP
P
I DP
n
i
D
i
;
ei
DP
D
e
Pn
?
e
P
?
P
I
e
i
P
?
?
e
P
I
:::
e
P
e
e
Pn
:::
?
n
?
I DP
P
DP
; : : :; n
e
?
P
e
Pn
Pi
?
Pi
I
I
P
D; ?
I
e
:::
D; ?
Pn
D; ?
D; ?
P
P
e
?
D
:::
e
e
Pn
?
Pn
?
i
D; ?
P
Pi
?
e
:::
?
D; ?
Pi
?
?
I
e
e
?
D; ?
Pi
D; ?
D
I
e
D
?
D
?
M
?
J
D
DM
I
?
?
D
?
I
I
Pi
Pi
?
?
D
P
?
J
e
P
e
D
M
e
: : : Pn
?
SEMISTAR DEDEKIND DOMAINS
25
of integers~ 1, = 1 2
. It follows that =
= ~ = ( 1 1 22 1
2
n
n
) = ( 1 2 ) (since ~ is a spectral semistar operation
dened by the set M( f )). Hence is a nite product of primes of . Therefore
is a local Dedekind domain [19, Theorem 37.8 ((1),(3))], that is,
is a
DVR.
Now we show that each quasi{~{prime ideal of is quasi{~{invertible.
Let be a
quasi{~{prime
of
and
let
0
6= 2 . Then, by assumption, ( )~ = ( 1 1 2 2 n )~ , with 1 2
nonzero prime ideals of and 1, = 1 2
.
Since is obviously invertible (and thus, clearly, quasi{~{invertible), then each
is quasi{~{invertible [16, Lemma 2.10]. Moreover, since is a quasi{~{ideal of
, then 1 1 2 2 n ( 1 1 2 2 n )~ \ . Therefore, for
some , with 1 , and since is a DVR, we have = . Hence is a
quasi{~{invertible ideal of . Therefore, by Theorem 4.9, we conclude that is
~{Dedekind.
Remark 4.26. It is clear that, if
is a {DD then, for each nonzero ideal of ,
such that f 6= f , we have a unique factorization f = ( 1 1 2 2 n ) f ,
for some family of quasi{ f {prime ideals , with f 6= f , and for some family
of positive integers , = 1 2
, since ~ = f (Proposition 1.6). The converse
is not true. For instance, take , and as in Remark 2.3(2). For each nonzero
proper ideal of , we have f = = = ( ) f , for some positive integer
, since is a DVR. Note that this representation is unique, since is local with
maximal ideal and dim( ) = 1. But we have already observed that is not a
{DD.
Next result generalizes to the semistar setting [19, Theorem 38.5 ((1),(3))].
ei
:::
Pn
e
?
i
DM
;
P
e
P
; : : :; n
e
J
e
Pn
:::
?
DM
?
I DM
I DM
e
P
P
e
?
J
DM
DM
DM
?
?
D
D
e
: : : Pn
?
x
?
?
xD
P ; P ; : : : ; Pn
D
ei
xD
e
P
i
;
P
e
; : : :; n
?
Pi
?
D
Q
Q
P
e
Q
P
e
e
:::
Pn
j
n
j
P
e
e
P
e
:::
Pn
?
D
Pj
DQ
?
?
Q
Q
Q
Pj
Q
D
D
?
D
I
?
D
?
I
?
I
?
P
?
?
ei
i
Pi
;
; : : :; n
D
I
e
D
I
?
T
?
Pi
D
e
P
e
:::
D
?
e
Pn
?
?
?
IT
M
e
M
e ?
T
D
M
D
D
?
Let D be an integral domain which is not a eld and ? a semistar
operation on D. The following are equivalent:
(1) D is a ?{DD.
(2) For each
nonzero ideal I and for each a 2 I , a 6= 0, there exists b 2 I e? such
~
?
that I = ((a; b)D)~? .
Theorem 4.27.
(1) ) (2). We start by proving the following:
Claim. If
is a {DD, then the map 7! e establishes a bijection between
the set M( f ) (= M(e) by Proposition 1.4 (5)) of the quasi{ f {maximal ideals of
and the set M( ? ) of the ? {maximal ideals of (the Krull domain) e .
For each 2 M( f ), since e , it is easy to see that e =
\ e.
Therefore, e is a e_ {prime ideal of e and e \ = . Furthermore, by
Corollary 4.20, we know that e is a Krull domain and e_ = ? . On the other
hand, for each e_ {prime ideal of e, we know that \ is a quasi{e{prime of
[15, Lemma 2.3 (4)]). Since is a {DD (or, equivalently, a e{DD), we have
that each quasi{e{prime is a quasi{e{maximal (Proposition 2.2 (2)), thus we easily
conclude.
Let 2 , 6= 0, and f 1 2
g the (nite) set of quasi{ f {maximal
ideals such that 2 . Since i is a DVR, then i =
2 ,
i , for some
for each = 1 e2
. We use the fact that e is a Krull domain and, by the
Claim, that f ? = j 2 M( f )g is the dening family of the rank-one discrete valuation overrings of e, in order to apply the approximation theorem to e.
Let 1 2
be the valuations associated respectively to 1 2
n
and let be the valuation associated to
= e ? , for 0 2 M0 := M( f ) n
f 1 2
g. Set 1 := 1 ( 1) 2 := 2 ( 2)
:= ( ). Then there
exists 2 such that ( ) = , for each~ = 1 2 ~ , and ( ) 0, for each
0 2 M0 [19, Theorem 44.1]. We have = (( ) ) . Indeed, let 2 M( f ).
Proof.
D
?
?
M
?
tD e
D
M
M
tD e
?
?
D
D
?
DM
?
D
D
?
D
?
M
D
?
N
?
Mi
D
DM
M
K
?
xi DM
xi
I
?
?
?
D
?
DM ; DM ; : : : ; DM
vM 0
b
?
?
I DM
v ; v ; : : : ; vn
M
D
?
DM
D
M ; M ; : : : ; Mn
M DM
tD e
D
?
;::: ;n
Me
?
M
M ; M ; : : : ; Mn
;
D
?
?
?
a
a
i
?
?
D
I
M
?
N
D
a
?
M
?
DM 0
k
vi b
v
x
;k
v
ki
i
I
?
D
x
;
?
M0e
; : : : ; kn
;::: ;n
a; b D
?
M
?
vn xn
vM 0 b
M
?
26
If = , for some , then
=
) i . If
i =
i =
i = (
6= for each , then
= =( ) .
(2) ) (1). Let 2 M( f ) and a nonzero ideal of . Let 2 , 6= 0, there
exists 2 , 2 , such that 2 := \ . Then, by assumption,~ there
exists 2 ~ e such that ~ = (( ) )~ . Therefore, we have =
=
=
(( ) ) = ( ) = ( ) . By [19, Theorem 38.5], is a Dedekind
domain, and hence a DVR. Thus, is a {ADD, hence, in particular, is is a P~MD
(Corollary 2.6 and Proposition 2.2(2)). In addition, from the assumption and from
[16, Lemma 2.3], we deduce that is ~{Noetherian (Lemma 3.3), hence is a
{DD (Corollary 4.3 and Proposition 4.1(1)).
Remark 4.28. Note that, if
is a {DD (and hence e = f ), then satises also
a statement concerning f , analogous to the statement (2) in Theorem 4.27:
(2f ) for each nonzero ideal of and for each 0 6= 2 , there exists 2 f such
that (( ) ) f = f .
But (2f ) does not imply that is a {DD. For instance, let
and be as in
Remark 2.3. Obviously, for each nonzero proper ideal of and for each nonzero
2 we have f = =
= ( ) = (( ) ) f , for some 1,
(where 2 f \ ), but is not a {DD.
References
M
M
Mi
i
Mi
I DM
i
I DM
M
s
D
b
I
sa; b D
?
?
?
DM
sa; b D
sa; b DM
bDM
DM
sa
I
xi DM
a; b DM
a; b DM
J
s = M
?
I DM
DM
I
J
a
?
J
a
?
I DM
a; b DM
D
J
D
I DM
DM
?
D
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?
D
?
D
?
?
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D
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I
a; b D
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a
I
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D; T
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a
I
D
X
n
I
I
?
?
D
b
I
?
?
IT
D
X
n
T
a; X
n
T
a; X
?
D
n
D
?
n
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