Nagata Transforms and Localizing Systems
Marco Fontana
Nicolae Popescu
Dipartimento di Matematica
Universita degli Studi, Roma Tre
00146 Roma, Italia
e{mail: [email protected]
Abstract.
Institut de Mathematiques
Academie de Roumanie
7070 Bucarest, Roumanie
e{mail: [email protected]
The main purpose of this paper is to undertake a more in-depth study
of the link between Nagata ideal transforms and the rings of fractions with respect
to localizing systems. The principal result obtained here is a characterization of
when the Nagata ideal transform T (I ) of an ideal I of an integral domain R is a
ring of fractions of R with respect to a localizing system; in this case, we give also a
description of the localizing system
F
of R such that T (I ) = RF .
1. Introduction and preliminary results
Let R be an integral domain with quotient eld K . For each ideal I of R, the
following overring of R
(1:1)
TR (I ) = T (I ) :=
[ (R : In) = fz 2 K : (R :R zR) In for some n 0g
n0
is called the Nagata (ideal) transform of R with respect to I .
Ideal transforms are a very useful tool in various aspects of commutative ring
theory. A particularly important application was the treatment of Hilbert's XIV
Problem given by Nagata in [N1], [N2] and [N3]. Nagata transforms were also
employed in the study of the catenary chain conjectures (cf. [R1] and [R2]) and
Brewer, Brewer{Gilmer and Hedstrom have proved that Nagata transforms are very
th
1
2
useful in the study of the overrings of an integral domain (cf. [Br], [BrG], [He1]
and [He2]).
Other interesting applications of the ideal transforms are mentioned in [F], where
an ample bibliography is listed.
Outside the Noetherian setting, the behaviour of Nagata transform is not entirely
satisfactory [F]. Kaplansky in [K] considered, for each ideal I of R, the following
overring of R:
(1:2)
R(I ) = (I ) := fz 2 K : rad((R :R zR)) I g
called (in [F]) the Kaplansky (ideal) transform of R with respect to I . It is straightforward that, for each ideal I , T (I ) (I ) and, if I is nitely generated, then
T (I ) = (I ).
A generalized multiplicative system (or a multiplicative system of ideals ) S of R
is a multiplicatively closed set of ideals of R. The following overring of R:
RS := fx 2 K : xI R for some I 2 Sg
is called the generalized transform or the (generalized) ring of fractions of R with
respect to S (cf. [HOP], [H], [AB], [BS]). This terminology is justied by the fact
that if N(I ) := fI n : n 0g then N(I ) is obviously a generalized multiplicative
system of R and
(1:3)
T (I ) = RN I ;
and by the fact that the ring of fractions S R of R with respect to a multiplicative
system S of R coincides with RS , where S = fsR j s 2 Sg.
Note that if S is a generalized multiplicative system of R then also
S := fJ : J ideal of R such that J I for some I 2 Sg
is a generalized multiplicative system called the saturation of S and RS = RS .
A generalized multiplicative system S is saturated if S = S .
A distinguished class of generalized multiplicative systems is given by the localizing (or topologizing) systems of ideals introduced by Gabriel [Ga] (cf. also [B, p.
157], [P], [C] and [St]). We recall that a localizing system of ideals F of R is a set
of ideals of R verifying the following conditions:
(LS1)
I 2 F and I J ) J 2 F ;
( )
1
ideal of R such that (J :R iR) 2 F for each i 2 I ) J 2 F :
For instance, for each subset of prime ideals of R,
F () := fI : I ideal of R such that I 6 P for each P 2 g
is a localizing system of R. If P is a prime ideal of R, we denote simply by F (P )
the localizing system F (fP g). It is obvious that:
F () = \fF (P ) : P 2 g :
(LS2)
I 2F ; J
3
A localizing system of nite type is a localizing system F such that for each I 2 F
there exists a nitely generated ideal J 2 F with J I . It is well known that if F
is a localizing system of nite type then F = F (), where := fP : P prime ideal
of R and P 2= Fg [FHP, Lemma 5.1.5].
The converse is not true in general, since F is a localizing system of nite type
if and only if F = F () where is a quasi{compact subspace of Spec(R) [FHP,
Proposition 5.1.8].
Since a localizing system F is a multiplicatively closed and saturated system of
ideals [FHP, Proposition 5.1.1], we can consider the generalized ring of fractions
RF of R with respect to F . Note that it is possible to nd an overring of an
integral domain R which is a ring of fractions of R with respect to a generalized
multiplicative system of ideals but is such that none of the multiplicative systems
giving rise to this overring is a localizing system [F].
If F = F () for some subset of prime ideals of R, then it is well known that
RF = \fRP : P 2 g
[FHP, Proposition 5.1.4].
For each ideal I of R, let
D(I ) := fP 2 Spec(R) : P 6 I g
K = K(I ) := F (D(I )) = fJ ideal of R : P 2 Spec(R) ; J P ) I P g;
then Kaplansky [K] (and Hays [Ha, Theorem 1.7]) proved that
(1:4)
(I ) = RK I :
Since, when I is nitely generated, T (I ) = (I ), the equality (1.4) generalizes a
result obtained by Brewer [Br, Theorem 1.5]. Furthermore, (1.4) provides also an
extension of the following representation given by Nagata [N3, Lemma 2.4]: if I is
a nonzero ideal of a Krull domain R, then
(1:5)
TR (I ) = \fRQ : Q 2 Spec(R); Q 6 I and ht(Q) = 1g =
= \fRP : P 2 Spec(R); P 6 I g = RK I :
In this situation, the ring TR (I ), which is always by (1.3) a generalized ring of
fractions of R with respect to the multiplicatively closed set of ideals N(I ), is also
a ring of fractions of R with respect to a localizing system (i.e. K(I )).
Note that the inclusion T (I ) (I ), that holds in general, may be also deduced
from the inclusion of the system of ideals N(I ) K(I ). This inclusion can be
proved synthetically by the fact that:
(1:6)
J 2 N(I ) ) rad(I ) = rad(J \ I )
and by the following explicit description of K(I ):
(1:7)
K(I ) = fJ : J ideal of R such that rad(I ) = rad(J \ I )g
[FHP, Remark 5.8.5 (a)].
( )
( )
4
The main purpose of this paper is to undertake a more in-depth study of the link
between Nagata ideal transforms and rings of fractions with respect to localizing
systems. Indeed, one of the reasons that makes Kaplansky's generalization of the
ideal transform very satisfactory in the not necessarily Noetherian context is the
fact that (I ) = RF , where F is the localizing system K(I ) (cf. also [F] and
[FH]). The principal result obtained here is a characterization of when the Nagata
ideal transform T (I ) is a ring of fractions of R with respect to a localizing system;
in this case, we give also a description of the localizing system F of R such that
T (I ) = RF .
2. The Nagata localizing system
Let R be an integral domain and K its quotient eld.
We call the Nagata localizing system associated to an ideal I of R, and we
denote it by N (I ), the smallest localizing system of R containing I . Obviously
N (I ) contains the saturation of the multiplicatively closed system of ideals N(I )
considered in Section 1.
Proposition 2.1.
Let R be an integral domain. For each ideal I of R, we set
N0 (I ) := fJ : J ideal of R ; J I n for some n 1g
and, for each ordinal number , we dene by transnite induction a set of ideals
N (I ) of R in the following way:
{ if is not a limit ordinal, i.e. = + 1 for some ordinal number 0,
N (I ) := fJ : J ideal of R such that there exists J 2 N (I ) with (J :R xR)
N (I ), for each x 2 J g;
{ if is a limit ordinal,
2
N (I ) := [<N (I ) :
Then, for each ordinal number 0,
(a) The set N(I ) is a saturated multiplicative system of ideals of R;
(b) The Nagata localizing system N (I ) coincides with [N(I ).
Proof .
(a) We want to show that the set N(I ) veries the following properties:
(1) If 0, then
N (I ) N (I );
(2) J 2 0N(I ), J 0 an ideal0 of R with J J 0 ) J 0 2 N(I );
(3) J; J 2 N(I ) ) JJ 2 N(I ).
(1) is obvious, since if J 2 N (I ), then (J :R xR) = R, for each x 2 J 2 N (I ).
For (2) and (3) it is suÆcient to show the statements when = 0 and, by
transnite induction, when is not a limit ordinal (i.e. = +1, for some ordinal
number 0).
The case = 0 is obvious, since N (I ) is the saturation of the multiplicative
ideal system N(I ).
Assume that for a given ordinal number 0, N (I ) veries (2) and (3). It
is easy to see that property (2) is veried also by N (I ), by the denition of
N (I ).
For 1 k 2, let Jk ideal of N (I ) hence:
0
+1
+1
+1
5
there exist Jk; 2 N (I ) and
(Jk :R xk R) 2 N (I ), for each xk 2 Jk; .
Since
(J :R x R)(J :R x R) 2 N (I ) and (J :R x R)(J :R x R) (J J :R x x R);
then we have:
(J J :R x x R) 2 N (I ); for all x x 2 J ; J ; :
By the fact that the elements of the type x x generate J ; J ; , we can conclude
easily that J J 2 N (I ).
(b) Set N(I ) := [N(I ). Note that, for each ordinal number , N(I ) satises
(LS1) hence also N (I ) satises (LS1).
Furthermore, from the denitions, it follows easily that I 2 N(I ) N (I ), in
order to conclude we show that N(I ) satises also (LS2), i.e. N(I ) is a localizing
system of R.
Let L 2 N(I ) and let H be an ideal of R such that (H :R xR) 2 N(I ),
for each x 2 L. Therefore (H :R xR) 2 Nx (I ) for some ordinal number x
depending on x 2 L, where L 2 N (I ), for some ordinal number 0. Hence, if
:= supfx : x 2 L; g, then (H :R xR) 2 N (I ) for each x 2 L, with L 2 N (I ),
and thus H 2 N (I ) N(I ). 1
1
2
2
1
1 2
1
1
2
2
1 2
1 2
1 2
2
1 2
1
1
1 2
2
2
+1
+1
Proposition 2.2.
With the notation introduced above,
J 2 N (I ) ) rad(J \ I ) = rad(I ) :
Proof . The statement is obvious if J 2 N (I ) (cf. also (1.6)). By using Proposition 2.1 and transnite induction, it is suÆcient to show that if the statement
holds for each J 2 N (I ), when 0 , then the statement holds for each
J 2 N (I ). We can assume that J 2 N (I ) and J 6 I (because if I J
then J 2 N (I )). Then there exists J 2 N(I ) with (J :R xR) 2 N(I ) for each
x 2 J .
Let P be a prime ideal containing J \ I . Assume that P 6 I , hence P J . By
induction rad((J :R xR) \ I ) = rad(I ), for each x 2 J. Note that x(J :R xR) J
for each x 2 J, hence also x((J :R xR) \ I ) J P for each x 2 J \ I . Two
cases are possible.
Case 1: (J :R xR) \ I P , for some x 2 J \ I . In this case I P because
rad((J :R xR) \ I ) = rad(I ) and, hence, we reach a contradiction.
Case 2: x 2 P , for each x 2 J \ I . Since J 2 N (I ), then rad(J \ I ) = rad(I ).
In this situation, we have that P J \ I hence P I : a contradiction.
We conclude that the only possible situation is the following: if a prime ideal
contains J \ I then it contains I , hence rad(J \ I ) = rad(I ). Remark 2.3. Recall that we have already noticed (cf. (1.4) and (1.7)) that the
Kaplansky transform R(I ) is the ring of fractions with respect to the localizing
system K(I ) = fJ : J ideal of R such that rad(J \ I ) = rad(I )g. By the previous
0
+1
+1
0
6
proposition we have N (I ) K(I ) and hence we (re)obtain easily that TR (I ) RN I RK I = R (I ).
For each ordinal number , we can dene by transnite induction an overring of
the Nagata transform TR (I ) in the following way:
T ;R (I ) := TR (I )
{ if is not a limit ordinal, i.e. = + 1 for some ordinal number 0, then
T;R (I ) := TT;R I (IT;R (I ))
{ if is a limit ordinal, then
T;R (I ) := [<T;R (I ) :
If no ambiguity occurs, we will denote simply by T the ring T;R(I ), thus
T = T (IT ).
For each ordinal number , we can consider the following subring of K
RN I := fx 2 K : (R :R xR) 2 N (I )g ;
since, for each ordinal number 0, N(I ) is a (saturated) multiplicative system
of ideals of R (Proposition 2.1 (a)).
( )
( )
0
( )
+1
( )
Lemma 2.4.
(a)
(b)
(c)
(d)
With the notation introduced above, then
TR (I ) = RN0 (I ) ;
T1;R (I ) = RN1 (I ) ;
for each ordinal number 2, T;R (I ) RN (I );
RN (I ) = [RN (I ) .
Proof . (a). It is a clear since N (I ) is the (multiplicative system of ideals of R)
saturation of N (I ) considered in Section 1, and thus TR (I ) = RN I = RN I .
(b). We denote simply by T the domain T;R(I ), for each ordinal number 0.
Let y 2 T , then y(IT )n T and thus yI n T for some n 0.
Set J := (R :R yR). For each x 2 I n, we have yx 2 T , whence (J :R xR) =
(R :R xyR) 2 N (I ) (by (a)). Therefore J = (R :R yR) 2 N (R). Conversely, let
y 2 K be such that (R :R yR) 2 N (I ). We know that there exists n 0 such that,
for each x 2 I n, ((R :R yR) :R xR) 2 N (IN). In particular, there exists
Nx 0
(depending non x) such that (R :R yxR) I x , hence yx 2 T , i.e. yI n T , and
thus y(IT ) T . This fact implies that y 2 T .
(c) is proved by transnite induction. To avoid the trivial case, we can assume
that is not a limit ordinal, then = + 1, for some ordinal number 0,
and thenproof is analogous to that of (b). More precisely, let y 2 T = T
then yI T T for some n 0. Set J := (R :R yR). Since yx 2 T for
each x 2 I n, then using the inductive hypothesis (i.e. T RN I ) we have
(J :R xR) = (R :R yxR) 2 N (I ). Since x varies in I n and I n 2 N (I ) N (I )
for each 0, then J 2 N (I ) = N(I ), thus y 2 RN I .
0
( )
1
0
0
0( )
0
0
0
1
1
0
0
0
0
0
1
+1
( )
0
+1
( )
7
(d). It is obvious that [RN I RN I . Conversely, if x 2 RN I then there
exists an ordinal number and an ideal J 2 N(I ) such that J (R :R xR).
Therefore (R :R xR) 2 N(I ) and hence x 2 RN I . ( )
( )
( )
( )
Theorem 2.5. Let I be an ideal of an integral domain R. The following are
equivalent:
(i) there exists a localizing system F of R such that TR (I ) = RF ;
(ii) TR(I ) = RN (I);
(iii) TR(I ) = T1;R(I ).
Proof . (i) ) (ii). The set LSR(TR (I )) := fG : G localizing system of R such
that RG = TR (I )g ordered under set{theoretic inclusion has a rst element G [FHP,
Lemma 5.1.19]. More precisely, G = [G where G := fJ : J ideal of R such that
J (R :R yR) for some y 2 TR (I )g and if is not a limit ordinal, i.e. = + 1
for some ordinal , then
G := fJ ideal of R such that there exists an ideal J 0 of R and J 00 2 G with J J 0
and (J 0 :R y00R) 2 G for each y00 2 J 00g;
if is a limit ordinal, then
G := [<G :
Note that if y 2 TR (I ), then y 2 (R : I ny ) for some ny 0, thus (R :R yR) I ny .
From this fact we deduce that J 2 G implies that J contains a power of the ideal
I , hence G N (I ). Therefore, by transnite induction and by Proposition 2.1, it
follows that G N (I ). We can conclude that G = N (I ) by the minimalityproperty
of N (I ), thus TR(I ) = RG = RN I .
(ii) ) (iii) follows from Lemma 2.4, since T T RN I .
(iii) ) (i). For each ordinal number , we denote simply by T the ring T;R(I ).
In the present situation, for each ordinal number 0, obviously we have
T = T :
Furthermore, we claim that, for each ordinal number 0,
(2:5:1)
T = RN I = [f(RN I : J ) : J 2 N(I )g :
For = 0 (or = 1) the rst equality in (2.5.1) holds in general by Lemma 2.4
(a) (or (b)) (without the assumption T = T ).
It is obvious, in general, that
RN I [f(RN I : J ) : J 2 N (I )g
since, by denition, x 2 RN I is equivalent to the fact that J := (R :R xR) 2
N (I ) and thus x 2 (R : (R :R xR)) (RN I : J ).
In order to show that the second equality in (2.5.1) holds when = 0, we take
x 2 K such that xJ TR (I ), for some J 2 N (I ), and we want to prove that
x 2 TR (I ). Without loss of generality we can assume that J = I n , for some n 1.
For each y 2 I n we have xy 2 TR (I ) and, thus, xyI my R for some my 0.
Therefore ((R :R xR) :R yR) = (R :R xyR) I my , hence (R :R xR) 2 N (I ) and
thus we conclude that x 2 T ;R(I ) = TR(I ).
0
0
0
0
( )
0
1
( )
0
( )
( )
0
( )
1
( )
( )
( )
0
1
1
8
Let 1. We can assume by transnite induction that (2.5.1) holds for each
ordinal number < .
If is a limit ordinal then, by the denitions, it is obvious that (2.5.1) holds
also for .
We assume that = + 1, for some ordinal number 0, and we want to
prove that the rst equality in (2.5.1) holds.
Let x 2 RN I , hence (R :R xR) 2 N(I ) and, to avoid the trivial case, we can
assume that (R :R xR) 62 N (I ). This means that there exists an ideal J 2 N (I )
such that, for each y 2 J , we have (R :R xyR) = ((R :R xR) :R yR) 2 N (I ).
By the inductive hypothesis, xy 2 T = RN I for each y 2 J . Therefore x 2
(RN I : J ) with J 2 N (I ) and thus, also by the inductive hypothesis, x 2
RN I = T T . By Lemma 2.4 (c) we conclude that the rst equality in (2.5.1)
holds, i.e. T = RN I . For the second equality in (2.5.1), let x 2 K be such that
xJ RN I for some J 2 N (I ) and, to avoid the trivial case, we can assume
that J 2 N (I ). By denition of N(I ), there exists an ideal H 2 N (I ) such
that (J :R hR) 2 N (I ), for each h 2 H . Since (xJ :R xhR) = (J :R hR), then
xh(xJ :R xhR) xJ T = T = RN I , thus by the inductive hypothesis
xh 2 RN I for each h 2 H . Therefore x 2 (RN I : H ) and, again by the
inductive hypothesis, (RN I : H ) RN I = T = T = RN I , hence we
deduce that x 2 RN I .
By Lemma 2.4 (d) and (2.5.1) we can conclude immediately that T = RN I . Example 2.6. Let k be a eld and fXn : n 1g a countable set of indeterminates over k . Let S be the additive monoid of all sequences of non negative
integers s = (sn : n 1). Set
Y Xnsn :
Xs :=
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
0
( )
n1
In the set M := fX : s 2 Sg we can dene in a natural way a multiplicative
monoid structure. Consider R := k[M] the monoid ring for M over k, then R is an
integral domain; let K be the quotient eld of R. Consider the following elements
of S:
u := (1; 2; : : : ; n 1; n; n + 1; : : :: : : )
and for each m 2,
u(m) := (1; 2; : : : ; m 1; 1; m + 1; : : :: : : ):
(For m = 1; u(1) = u.)
Let ym := Xu m and let I be the ideal of R generated by ym , for m 1.
It is not diÆcult to see that the element 1=Xm , which belongs to K , is inside
T = TR (I ), for all m 1.
If z := Xu, we claim that the element 1=z , hwhich is in K , does not belong to hT .
As a matter of fact if, for some h 1, (1=z)I R then, in particular, (1=z)yh
would belong to R. But
1 yh = 1 Y X hn n
s
(
)
0
0
+1
z h+1
Xh+1
n1
n6=h+1
n
does not belong to R, hence we have a contradiction.
9
However, 1=z belongs to T = TTR I (ITR (I )) because, for each m 1, (1=z)ym
= 1=Xmm belongs to T , since -as we noticed before- 1=Xm 2 T and T is a ring.
We conclude that, in the present situation, T 6= T , hence, by Theorem 2.5 we
deduce that TR(I ) is not a ring of fractions of R with respect to some localizing
system.
1
1
( )
0
0
0
0
1
ACKNOWLEDGMENTS
M. Fontana was supported in part by the research funds of the Ministero
dell'Universita e della Ricerca Scientica e Tecnologica .
N. Popescu was supported by a visiting professorship of INdAM . Popescu thanks
the Dipartimento di Matematica of the Universita Roma Tre for its warm hospitality
during his visit in the spring 2000.
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10
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