# Ratliff-Rush Closure of Ideals in Integral Domains

###### Abstract.

This paper studies the Ratliff-Rush closure of ideals in integral domains. By definition, the Ratliff-Rush closure of an ideal of a domain is the ideal given by and an ideal is said to be a Ratliff-Rush ideal if . We completely characterize integrally closed domains in which every ideal is a Ratliff-Rush ideal and we give a complete description of the Ratliff-Rush closure of an ideal in a valuation domain.

###### Key words and phrases:

Ratliff-Rush closure, integral closure, Ratliff-Rush ideal, integrally closed ideal, reduction, Prüfer domain, valuation domain###### 2000 Mathematics Subject Classification:

Primary 13A15, 13A18, 13F05; Secondary 13G05, 13F30## 1. Introduction

Let be a commutative ring with identity and a regular ideal
of , that is, contains a nonzero divisor. The ideals of the
form
increase with . In the case where is a Noetherian ring, the
union of this family is an interesting ideal, first studied by
Ratliff and Rush in [22]. In [12], W. Heinzer, D. Lantz
and K. Shah called the ideal
the Ratliff-Rush closure of , or the Ratliff-Rush ideal
associated with . An ideal is said to be a Ratilff-Rush
ideal, or Ratliff-Rush closed, if . Among the
interesting facts of this ideal is that, for any regular ideal
in a Noetherian ring , there exists a positive integer such
that for all , , that is, all
sufficiently high powers of a regular ideal are Ratliff-Rush ideals,
and a regular ideal is always a reduction of its Ratliff-Rush
closure in the sense of Northcoot-Rees (see [17]), that is,
for some positive integer .
Also the ideal is always between and the integral
closure of , that is, ,
where satisfies an equation of the form , where for each . Therefore, integrally closed ideals, i. e.,
ideals such that , are Ratliff-Rush ideals. Since then, many
investigations of the Ratliff-Rush closure of ideals in a Noetherian
ring have been carried out, for instance, see [11],
[12], [16], [23] etc. The purpose of this paper is
to extend the notion of Ratliff-Rush closure of ideals to an
arbitrary integral domain and examine ring-theoretic properties of
this kind of closure. In the second section, we give an answer to a
question raised by B. Olberding [21] about the classification
of integral domains for which every ideal is a Ratliff-Rush ideal in
the context of integrally closed domains. This lead us to give a new
characterizations of Prüfer and strongly discrete Prüfer
domains. Specifically, we prove that “a domain is a Prüfer
(respectively strongly discrete Prüfer) domain if and only if
is integrally closed and each nonzero finitely generated
(respectively each nonzero) ideal of is a Ratliff-Rush ideal”
(Theorem 2.6). It turns that a Ratliff-Rush domain (i. e.,
domain such that each nonzero ideal is a Ratliff-Rush ideal) is a
quasi-Prüfer domain, that is, its integral closure is a Prüfer
domain. As an immediate consequence, we recover Heinzer-Lantz-Shah’s
results for Noetherian domains (Corollary 2.8). The third
section deals with valuation domains. Here, we give a complete
description of the Ratliff-Rush closure of a nonzero ideal in a
valuation domain (Proposition 3.2), and we state necessary
and sufficient condition under which the Ratliff-Rush closure
preserves inclusion (Proposition 3.3). We also extend the
Ratliff-Rush closure to arbitrary nonzero fractional ideals of a
domain , and we investigate its link to the notions of star
operations. We prove that “for a valuation domain , the
Ratliff-Rush closure is a star operation if and only if every
nonzero nonmaximal prime ideal of is not idempotent, and in this
case it coincides with the -closure” (Theorem 3.5).

Throughout, denotes an integral domain, its quotient field, and and its integral closure and complete integral closure respectively. For a nonzero (fractional) ideal of , the inverse of is given by . The -closure and -closure are defined respectively by and where ranges over the set of f. g. subideals of . We say that is divisorial (or a -ideal) if , and a -ideal if . Unreferenced material is standard as in [10] or [15].

## 2. Ratliff-Rush ideals in an integral domain

Let be an integral domain. A nonzero ideal of is
-stable (here stands for Lipman) if
. The ideal is stable (or
Sally-Vasconcelos stable) if is invertible in its endomorphisms
ring ([24]). A domain is -stable (respectively
stable) if every nonzero ideal of is -stable (respectively
stable). We recall that a stable domain is -stable [1, Lemma
2.1], and for recent developments on stability (in settings
different than originally considered), we refer the reader to
[1, 18, 19, 20]. We start this section with the following
definition which
extend the notion of Ratliff-Rush closure to nonzero integral ideals in an arbitrary integral domain.

###### Definition 2.1.

Let be an integral domain and a nonzero
integral ideal of . The Ratliff-Rush closure of is the
(integral) ideal of given by

. An integral
ideal of is said to be a Ratliff-Rush ideal, or Ratliff-Rush
closed, if , and is said to be a Ratliff-Rush
domain if each nonzero integral ideal of is a Ratliff-Rush
ideal.

The following useful lemma treats the Ratliff-Rush closure of some
particular classes of ideals.

###### Lemma 2.2.

Let be an integral domain. Then:

1-All stable (and thus all invertible) ideals are Ratliff-Rush.

2-If is a nonzero idempotent ideal of , then .

###### Proof.

1) Let be a stable ideal of and set .
Then . Now, let . Then and
for some positive integer . Composing
the two sides with and using the fact that , we
obtain . Iterating this process, we get
. Hence and therefore
, as desired.

2) Let be a nonzero idempotent ideal of . Then for each ,
. So . Hence
.

∎

The next proposition relates the Ratliff-Rush closure to the
-stability.

###### Proposition 2.3.

Let be an integral domain. If is a Ratliff-Rush domain, then is -stable.

###### Proof.

Assume that is a Ratliff-Rush domain. Let be a nonzero (integral) ideal of and let . Then there exists a positive integer such that . Let such that . Then implies that . Hence . Since , then (since is Ratliff-Rush) and so . Hence and therefore . So is -stable and therefore is -stable, as desired. ∎

It’s easy to see that for a finitely generated ideal of a domain
, in particular if is Noetherian, .
However, this is not the case for an arbitrary ideal of an integral
domain. Indeed, let be a valuation domain with maximal ideal
such that , and set . It is easy to
see that and (since all ideals of
a Prüfer domains are integrally closed). The next theorem establishes a
connection between stable domains, Ratliff-Rush domains and domains for which for all ideals .
For this, we need the following crucial lemma.

###### Lemma 2.4.

Let be an integral domain. If for every finitely generated ideal of , then is a Prüfer domain.

###### Proof.

Let be a maximal ideal of . To show that is a valuation domain, let , where . Let be the ideal of . Then . So . Thus for some and in . Dividing by , we get . By the , theorem ([15, Theorem 67]), we get that either or , as desired. ∎

###### Theorem 2.5.

Let be an integral domain. Consider the following.

is stable.

is Ratliff-Rush.

for each nonzero ideal of .

has no nonzero idempotent prime ideals.

Then .
Moreover, If is a semilocal Prüfer domain, then
.

###### Proof.

We are now ready to announce the main theorem of this section. It
gives a classification of the integral domains for which every ideal
is a Ratliff-Rush ideal in the context of integrally closed domains
and states a new characterization of Prüfer and strongly discrete
Prüfer domains. Recall that a Prüfer domain is said to be strongly
discrete if for each nonzero prime ideal of .

###### Theorem 2.6.

Let be an integrally closed domain. The following
statements are equivalent.

for every finitely generated (respectively
every) nonzero ideal of .

is Prüfer (respectively strongly discrete Prüfer).

###### Proof.

By
Lemma 2.4, is a Prüfer
domain. Moreover, if each ideal is a Ratliff-Rush ideal, by Theorem 2.5, is strongly discrete.

. Let be a Prüfer domain. Then every
finitely generated ideal is invertible and therefore a Ratliff-Rush
ideal by Lemma 2.2. Assume that is a strongly discrete
Prüfer domain. Let be a nonzero ideal of and let . Then and for some
positive integer . Let be a maximal ideal of . If
, then . Assume that
. Since and , then . Since is strongly
discrete, then is a strongly discrete valuation domain. By
Theorem 2.5, . Hence . So . Hence
, as desired.

∎

The following example shows that the above Theorem is not true if
is not integrally closed.

###### Example 2.7.

Let be the field of rational numbers, an indeterminate over and . Set . Then is stable. Indeed, Let be a nonzero (integral) ideal of . Since is local with maximal ideal , then . If is an ideal of , then for some . If is not an ideal of , then , where is a -vector space. Since , then and so . Therefore is stable and then Ratliff-Rush by Theorem 2.5. However, is not a Prüfer domain ([4, Theorem 2.1]).

Our next corollary recovers Heinzer-Lantz-Shah’s results for
Noetherian domains.

###### Corollary 2.8.

(cf. [12, Proposition 3.1 and Theorem 3.9] Let be a Noetherian domain. Then is a Ratliff-Rush domain if and only if is stable.

###### Proof.

We recall that a domain is said to be strong Mori if
satisfies the ascending chain conditions on -ideals [7].
Trivially, a Noetherian domain is strong Mori and a strong Mori
domain is Mori. The next corollary shows that the
Ratliff-Rush property forces a strong Mori domain to be Noetherian.

###### Corollary 2.9.

Let be a strong Mori domain. If is a Ratliff-Rush domain, then is Noetherian.

###### Proof.

By Lemma 2.4, is a Prüfer domain. Hence every maximal ideal of is divisorial ([5, Corollary 2.5] and [6, Theorem 2.6]). Now, let be a maximal ideal of . Since , then is Noetherian ([7, Theorem 3.9]). Hence is a Krull domain. But since is Prüfer, then so is . Hence is Dedekind and so . Then and therefore is Noetherian ([7, Corollary 3.10]). ∎

Recall that is seminormal if for each , implies that . Our next corollary states
some conditions under which a Ratliff-Rush Mori domain has dimension one.

###### Corollary 2.10.

Let be a Mori domain such that either or is seminormal. If is a Ratliff-Rush domain, then .

###### Proof.

Assume that is a Ratliff-Rush domain. By
Lemma 2.4, is a Prüfer domain.

(1) If , then is a Krull
domain ([2, Corollary 18]). Since ,
then is a Prüfer domain, and therefore Dedekind.
Hence . By [3, Corollary 3.4],
, as desired.

(2) Assume that is seminormal. If , then has a
maximal ideal such that . Set
. Since is a local Mori
domain which is seminormal and , then
contains a nondivisorial maximal ideal contracting to
([3, Lemma 2.5]). Since is a Prüfer domain
(Lemma 2.4) and combining [5, Corollary 2.5]) and
[6, Theorem 2.6], we get that every maximal ideal of is
a -ideal and so a -ideal since is Mori, which is absurd.
Hence , as desired.
∎

## 3. Ratliff-Rush ideals in a Valuation domain

It’s well-known that the maximal ideal of a valuation domain
is either principal or idempotent, any nonzero prime ideal of
is a divided prime ideal, that is, , and any
idempotent ideal is a prime ideal. Also we recall that a valuation
domain is a domain, that is, for each nonzero ideal of ,
either or is a prime ideal of
([8, Proposition 2.1]), and for each positive integer ,
([13, Remark 2.13(b)]). We will often use
this facts without explicit mention. Finally is strongly
discrete if it has no nonzero idempotent prime ideal ([9, chapter
5.3]).

###### Lemma 3.1.

Let be a valuation domain, a nonzero ideal of and assume that . Then .

###### Proof.

Let be a nonzero ideal of and assume that
. If , then by
Lemma 2.2 and therefore .
Assume that is a prime ideal of . Since is a
valuation domain, then is -stable. So
for each positive integer . Let and . Then and for some
positive integer . Since , then
. Hence . To show that , it suffices to
prove that . Suppose that . Then . Since , then .
So and for some
positive integer . Hence (since ) and therefore
. Hence and therefore is an
idempotent prime ideal of . By Lemma 2.2,
, which is absurd. Hence . So and then . Hence and therefore .

∎

The next proposition describes the Ratliff-Rush closure of a nonzero
integral ideal in a valuation domain.

###### Proposition 3.2.

Let be a nonzero integral
ideal of a valuation domain . Then:

(1) if and only if is an idempotent prime ideal of .

(2) Assume that . Then either ,
or for some nonzero prime ideal of .

###### Proof.

(1) If is an idempotent prime ideal of , by
Lemma 2.2, . Conversely, assume that
. Then there exists a positive integer such that
. Hence . By induction,
. So is an idempotent ideal of . Hence
is a prime ideal of . Then and therefore , as
desired.

(2) Assume that . If , then
by Lemma 2.2. Assume that
is a prime ideal. Then and for
each positive integer , since is a
domain. Let . Then and for some positive integer . So . Hence and therefore
. Now, assume that .

To complete the proof, we will show that .
Since
(Lemma 3.1), then is an ideal of .
Suppose that . Let . Since is a valuation domain,
then . So . Hence is a proper ideal of
. So ( is the maximal
ideal of ). Hence , a contradiction. It follows that
, as desired.
∎

Our next proposition shows that the Ratliff-Rush closure of an ideal
in a valuation domain is itself a Ratliff-Rush ideal, and gives
necessary and sufficient condition for preserving the Ratliff-Rush
closure under inclusion.

###### Proposition 3.3.

Let be a nonzero ideal of a
valuation domain . Then

1) .

2) for every ideals if
and only each nonzero nonmaximal prime ideal of in not
idempotent.

###### Proof.

1) If or , then clearly
. Assume that . By Proposition 3.2,
where is a prime ideal of
(note that , otherwise , by
Lemma 2.2). For simplicity, we set . Our aim
is to prove that . If , then
by Lemma 2.2. Assume that . By
Lemma 3.1, , where . So
. Let . Then and
for some positive integer . Composing
the two sides with and using the fact that
, we obtain . Hence
. Now, if
, then let . Since is a
valuation domain, then . So .
Hence ([14]). So
. Then , a contradiction. Hence . So
. Hence
, as desired.

2) Assume that for every ideals
. Suppose that there is a nonzero nonmaximal prime
ideal of such that . Let , where
is the maximal ideal of . Since is a valuation domain,
then . By Lemma 2.2 and the hypothesis,
, which is absurd.

Conversely, assume that each nonzero nonmaximal prime ideal of in not idempotent and let be ideals of . If , or , then clearly . If , by Proposition 3.2, is an idempotent prime ideal of . By the hypothesis, . So . Then and so . Hence we may assume that and . By Proposition 3.2, , where . Now, suppose that . Then let . Since is a valuation domain, then . So . Since is an ideal of , then . So . Therefore . If is nonmaximal, by the hypothesis, . Hence for some nonzero (since is the maximal ideal of ). Hence . So is stable and by Lemma 2.2, , which is absurd. Hence and . If is principal in , then so is and therefore , which is absurd. Hence . So . Let . Then . Hence . So . Hence , which is absurd. It follows that , as desired. ∎

Now, we extend the Ratliff-Rush closure to arbitrary nonzero
fractional ideals and we study its link to the notion of star
operations. Our motivation is [12, Example 1.11], which
provided an example of a Noetherian domain with a nonzero ideal
such that for some . First, we recall that a star operation on is a map , where denotes the
set of all nonzero fractional ideals of , with the following
properties for each and each :

and ;

and if , then
;

.

For more details on the notion of star operations, we refer the reader to [10].

###### Definition 3.4.

Let be an integral domain with
quotient field and let be a nonzero fractional ideal of
.

The generalized Ratliff-Rush closure of is defined by
, for some .
Clearly for any nonzero integral ideal
of .

It is easy to see that for a nonzero fractional ideal of a domain , is an -module which is a fractional ideal if . In particular if is conducive or -stable, then is always a fractional ideal of . The next theorem gives necessary and sufficient conditions for the generalized Ratliff-Rush closure to be a star operation on a valuation domain.

###### Theorem 3.5.

Let be a valuation domain. The generalized Ratllif-Rush closure on is a star operation if and only if each nonzero nonmaximal prime ideal of is not idempotent. In this case, it coincides with the -operation.

###### Proof.

Assume that the generalized Ratliff-Rush closure is a
star operation. Then, by Proposition 3.3, each nonzero
nonmaximal prime ideal of is not idempotent. Conversely, assume
that each nonzero nonmaximal prime ideal of is not idempotent.

Claim. For each integral ideal of ,
. Indeed, it suffices to show that
. If , then , as desired.
Assume that is a prime ideal of . Then .
Let . Then for some positive
integer . Since , we get . Now, if
, then . So . If
, by hypothesis, is not idempotent. Hence
(since is the maximal ideal of ). So
(here is an ideal of
). Hence and therefore , as desired.

Now, we prove the three properties of star operations.
Let and be nonzero fractional ideals of and .

(1) : if and only if for some positive integer , if and only if
, if and only if .

(2) : Let such that . By , Proposition 3.3(2) and the claim,
. Hence .

(3) : Clearly and by and
Proposition 3.3(1), .

To complete the proof, we prove that for each
nonzero fractional ideal of . Since the -operation is the
largest star operation on , then .
Suppose that for some ideal of .
Then is not divisorial in . Hence for some and . Since is idempotent, then is not
divisorial. So . Hence (note that
by and Lemma 2.2
), which is absurd.
∎

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