Star Stability and Star Regularity for Mori Domains
aa r X i v : . [ m a t h . A C ] J u l STAR STABILITY AND STAR REGULARITY FOR MORI DOMAINS
STEFANIA GABELLI AND GIAMPAOLO PICOZZA
Abstract.
In the last few years, the concepts of stability and Clifford regularity have beenfruitfully extended by using star operations. In this paper we study and put in relationthese properties for Noetherian and Mori domains, substantially improving several resultspresent in the literature.
Introduction
In this paper, we study stability and Clifford regularity of Noetherian and Mori domainswith respect to star operations, substantially improving several results present in the litera-ture and answering some questions left open.Recall that a domain R is stable if each nonzero ideal I of R is invertible in its endo-morphism ring E ( I ) := ( I : I ). Stable domains have been thoroughly investigated by B.Olberding [26, 27, 28, 29].Stability implies Clifford regularity. A domain is called Clifford regular if each nonzeroideal I of R is von Neumann regular , that is I = I J for some ideal J . This conceptwas studied for orders in quadratic fields by P. Zanardo and U. Zannier in [34]. Later S.Bazzoni and L. Salce proved that all valuation domains are Clifford regular [6] and S. Bazzonideepened the study of Clifford regularity in [2, 3, 4, 5].Stability with respect to semistar operations was introduced and studied by the authorsof this paper in [15].The first attempt to extend the notion of Clifford regularity in the setting of star operationsis due to S. Kabbaj and A. Mimouni, who studied Clifford t -regularity for P v MDs and Moridomains [20, 21, 22, 23]. Successively F. Halter-Koch, in the language of ideal systems,introduced Clifford ∗ -regularity for any star operation of finite type [18] and the authors ofthis paper continued this study in [16].Here we mainly consider the cases where ∗ = d, w, t . In fact the most interesting resultson star stability and star regularity were obtained in [15] and [16] for star operations spectraland of finite type. In addition, if ∗ is spectral and of finite type, ∗ -regularity implies ∗ = w [16, Corollary 1.7]; in particular, if R is Clifford regular, then w = d .Definitions are given in Section 1.In Section 2, we prove that t -regularity and t -stability extend to t -compatible Mori over-rings (Proposition 2.1). We also show that for Mori domains t -regularity, t -stability and w -stability are t -local properties, while w -regularity is a t -local property for strong Moridomains (Proposition 2.5). Key words and phrases.
Stability, Clifford regularity, star operation, Mori domain.
In Section 3, we prove that Noetherian ∗ -stable domains have ∗ -dimension one and strongMori t -stable domains have t -dimension one (Corollary 3.3), extending the well known resultthat Notherian stable domains are one-dimensional and a result of Kabbaj and Mimouni[23, Lemma 2.7]. This allows us to show that, for strong Mori domains, w -regularity and w -stability are equivalent (Corollary 3.11), while t -regularity and t -stability are equivalentonly in dimension one (Corollary 3.10). In relation to a question of Kabbaj and Mimouni [23,Question 2.11(3)], we also prove that the w -integral closure of a strong Mori Boole w -regulardomain is a unique factorization domain (Proposition 3.12).In Section 4, we consider the Mori case. We are unable to extend all the results obtainedfor the t -operation in the Noetherian case; however we show that w -stable Mori domainshave t -dimension one (Proposition 4.1) and that in t -dimension one w -regularity and w -stability are equivalent (Proposition 4.3). As a consequence, we are able to give an answerto a question posed by Kabbaj and Mimouni in [20, page 633] (Corollary 4.4): namely weshow that a Mori w -regular domain has t -dimension one if and only if its w -integral closureis a Krull domain (Corollary 4.4).1. Preliminaries and notation
Throughout all the paper, R will be an integral domain and K its field of fractions. If I is a nonzero fractional ideal of R , we call I simply an ideal and if I ⊆ R we say that I is an integral ideal .We assume that the reader is familiar with the properties of star operations (see for exam-ple [17, Sections 32, 34]). Occasionally we will also consider semistar operations; standardmaterial about semistar operations can be found in [10]. We just recall some basic notionsthat will be used in the paper.By F ( R ) we denote the set of nonzero R -submodules of K and by F ( R ) the semigroupof all ideals of R . A semistar operation (respectively, a star operation ) ∗ on R is a map F ( R ) → F ( R ) (respectively, F ( R ) → F ( R )), I I ∗ , such that the following conditionshold for each 0 = a ∈ K and for each I , J ∈ F ( R ) (respectively, F ( R )):(i) ( aI ) ∗ = aI ∗ (respectively, ( aI ) ∗ = aI ∗ and R = R ∗ );(ii) I ⊆ I ∗ , and I ⊆ J ⇒ I ∗ ⊆ J ∗ ;(iii) I ∗∗ = I ∗ .If ∗ is a semistar operation on R such that R ∗ = R , ∗ is called a (semi)star operation on R and its restriction to the set of ideals F ( R ) is a star operation on R , still denoted by ∗ .Conversely, any star operation ∗ on R can be extended to a (semi)star operation by setting I ∗ = K for all I ∈ F ( R ) \ F ( R ).We will be mainly concerned with star operations and (semi)star operations.If ∗ is a semistar operation on R and D is an overring of R , the restriction of ∗ to the setof D -submodules of K is a semistar operation on D , here denoted by ∗ | D or by ˙ ∗ when noconfusion arises. When D ∗ = D , ˙ ∗ is a (semi)star operation on D [13, Proposition 2.8]. Notethat ˙ ∗ shares many properties with ∗ (see for instance [31, Proposition 3.1]); for example, if ∗ is of finite type then ˙ ∗ is of finite type [13, Proposition 2.8]. TAR STABILITY AND STAR REGULARITY FOR MORI DOMAINS 3
To any semistar operation ∗ , we can associate a semistar operation of finite type ∗ f , definedby I ∗ f := S { J ∗ | J ∈ F ( R ) finitely generated and J ⊆ I } , and a semistar operation spectraland of finite type e ∗ , defined by I e ∗ := T M ∈∗ f -Max( R ) IR M , for all I ∈ F ( R ).If ∗ is a (semi)star operation, an ideal I is a ∗ -ideal if I = I ∗ and I is called ∗ -finite (or of finite type ) if I ∗ = J ∗ = J ∗ f for some finitely generated ideal J ∈ F ( R ).A ∗ -prime ideal is a prime ideal which is also a ∗ -ideal and a ∗ -maximal ideal is a ∗ -idealmaximal in the set of proper integral ∗ -ideals of R . We denote by ∗ -Spec( R ) (respectively, ∗ -Max( R )) the set of ∗ -prime (respectively, ∗ -maximal) ideals of R . If ∗ is a (semi)staroperation of finite type, by Zorn’s lemma each ∗ -ideal is contained in a ∗ -maximal ideal,which is prime. In this case, R = T M ∈∗ -Max( R ) R M . We say that R has ∗ -finite character ifeach nonzero element of R is contained in at most finitely many ∗ -maximal ideals.When ∗ is of finite type, a minimal prime of a ∗ -ideal is a ∗ -prime. In particular, anyminimal prime over a nonzero principal ideal (in particular any height-one prime) is a ∗ -prime, for any (semi)star operation ∗ of finite type. The ∗ -dimension of R is the supremumof the lengths of the chains of prime ideals (0) ⊆ P ⊆ . . . ⊆ P n ⊆ . . . , where P i ∈ ∗ -Spec( R ).The identity is a (semi)star operation denoted by d , I d := I for each I ∈ F ( R ). Twonontrivial (semi)star operations which have been intensively studied in the literature are the v -operation and the t -operation. The v -closure of I is defined by setting I v := ( R : ( R : I )),where for any I, J ∈ F ( R ) we set ( J : I ) := { x ∈ K : xI ⊆ J } . A v -ideal of R is alsocalled a divisorial ideal . The t -operation is the (semi)star operation of finite type associatedto v and is therefore defined by setting I t := S { J v | J finitely generated and J ⊆ I } . The(semi)star operation spectral and of finite type associated to v is called the w -operation andis defined by setting I w := T M ∈ t -Max( R ) IR M .If ∗ and ∗ are (semi)star operations on R , we say that ∗ ≤ ∗ if I ∗ ⊆ I ∗ , for each I ∈ F ( R ). This is equivalent to the condition that ( I ∗ ) ∗ = ( I ∗ ) ∗ = I ∗ . If ∗ ≤ ∗ , then( ∗ ) f ≤ ( ∗ ) f and e ∗ ≤ e ∗ . Also, for each (semi)star operation ∗ , we have d ≤ ∗ ≤ v (so that ∗ f ≤ t and e ∗ ≤ w ) and e ∗ ≤ ∗ f ≤ ∗ (so that w ≤ t ≤ v ).For any (semi)star operation ∗ , the set of ∗ -ideals of R , denoted by F ∗ ( R ), is a semigroupunder the ∗ -multiplication, defined by ( I, J ) ( IJ ) ∗ , with unit R . We say that an ideal I ∈ F ( R ) is ∗ -invertible if I ∗ is invertible in F ∗ ( R ), equivalently ( I ( R : I )) ∗ = R . If ∗ is a(semi)star operation of finite type, then I is ∗ -invertible if and only if I is ∗ -finite and I ∗ R M is principal for each M ∈ ∗ -Max( R ) [24, Proposition 2.6].For a semistar operation ∗ on R , if I is an ideal of R and E := E ( I ∗ ) := ( I ∗ : I ∗ ), itis easy to see that E ∗ = E . Thus the restriction of ∗ to the set of E -submodules of K (denoted by ˙ ∗ := ∗ | E ) is a (semi)star operation on E . We say that an ideal I of R is ∗ -stable if I ∗ is ˙ ∗ -invertible in E and that R is ∗ -stable (respectively, finitely ∗ -stable ) if each ideal(respectively, each finitely generated ideal) of R is ∗ -stable. We also say that I is strongly ∗ -stable if I ∗ is principal in E and that R is strongly ∗ -stable if each ideal is strongly ∗ -stable.If ∗ is a star operation on R , denoting by P ( R ) the group of principal ideals of R , thequotient semigroup S ∗ ( R ) := F ∗ ( R ) / P ( R ) is called the ∗ -Class semigroup of R .We say that R is Clifford ∗ -regular , or simply ∗ -regular , if the semigroup S ∗ ( R ) is Cliffordregular. This means that each class [ I ∗ ] ∈ S ∗ ( R ) is (von Neumann) regular. Note thatthis is equivalent to saying that each ideal I ∗ is (von Neumann) regular in F ∗ ( R ), that is STEFANIA GABELLI AND GIAMPAOLO PICOZZA I ∗ = ( I J ) ∗ , for some nonzero ideal J of R ; in this case necessarily ( IJ ) ∗ = ( I ( E ( I ∗ ) : I ))) ∗ ,so that ∗ -stability implies ∗ -regularity. If [ I ∗ ] is regular, we say that I is ∗ -regular . If [ I ∗ ]is idempotent, that is [ I ∗ ] = [( I ) ∗ ] (equivalently ( I ) ∗ = xI ∗ for a nonzero x ∈ K ) we saythat I is Boole ∗ -regular and if each [ I ∗ ] is idempotent we say that R is Boole ∗ -regular .Clearly Boole ∗ -regularity implies Clifford ∗ -regularity. More precisely, we have the followingrelations. Proposition 1.1. [16, Proposition 1.5]
Let I be an ideal of R and, for a star operation ∗ on R , set E := E ( I ∗ ) . (1) If I is ∗ -stable, then I is ∗ -regular. Hence a ∗ -stable domain is Clifford ∗ -regular. (2) If I is ∗ -regular, then I ∗ is v E -invertible in E and if, in addition, I is finitely gener-ated, then I ∗ is t E -invertible in E (where v E and t E denote respectively the v -operationand the t -operation on E ). (3) I is strongly ∗ -stable if and only if I is Boole ∗ -regular and ∗ -stable. Hence a strongly ∗ -stable domain is precisely a Boole ∗ -regular ∗ -stable domain. Note that if ∗ ≤ ∗ , then ∗ -regularity (respectively, ∗ -stability) implies ∗ -regularity(respectively, ∗ -stability).For our purposes, it will be useful to work in the more general context of ∗ -Noetheriandomains. Given a star operation ∗ on R , R is called ∗ -Noetherian if it satisfies the ascendingchain condition on ∗ -ideals, or equivalently, if every ∗ -ideal is ∗ -finite. This implies that if R is ∗ -Noetherian then ∗ is of finite type. Note that if ∗ ≤ ∗ , then ∗ -Noetherianity implies ∗ -Noetherianity.Clearly when ∗ = d a ∗ -Noetherian domain is just a Noetherian domain. When ∗ = v or t , a ∗ -Noetherian domain is called a Mori domain and when ∗ = w it is called a strong Mori domain. We recall that R is a strong Mori domain if and only if it is a Mori domain suchthat R M is Noetherian for each M ∈ t -Max( R ). For the main properties of Mori domainswe refer to the survey article [1].For technical reasons, the most interesting results on star stability and star regularitywere obtained in [15] and [16] for star operations spectral and of finite type. In addition, weproved that, for ∗ of finite type, either ∗ -stability or e ∗ -regularity implies that e ∗ = w ([15,Corollary 1.6] and [16, Corollary 1.7]); in particular, if R is Clifford regular, then w = d .This implies that a Clifford regular strong Mori domain is indeed Noetherian.We observe that Kabbaj and Mimouni in [23] defined a t -stable domain as a domainwith the property that each t -ideal I is stable, that is invertible in its endomorphism ring E ( I ). However this condition is generally stronger than the usual definition, in fact anyNoetherian integrally closed domain R is t -stable (since each t -ideal I of R is t -invertible in R = ( I : I ) =: E ( I )), but a t -ideal of R need not be invertible [23, Example 2.9]. Howeverwe will show in Corollary 3.7 that the two definitions are equivalent in the Noetherian one-dimensional case. On the other hand, denoting by t E the t -operation on E := E ( I ), we havethat a t -regular t -ideal I of a Mori domain is always t E -invertible in E . This follows fromProposition 1.1(2), since I = J t , with J finitely generated. TAR STABILITY AND STAR REGULARITY FOR MORI DOMAINS 5 Overrings
Let R ⊆ D be an extension of domains. If ∗ denotes the w - or the t -operation on R and ∗ D denotes the respective operation on D , we say that the extension is ∗ -compatible if( ID ) ∗ D = ( I ∗ D ) ∗ D for each ideal (equivalently, finitely generated ideal) I of R . Also recallthat D is t -linked over R if ( Q ∩ R ) t ( R for each prime t D -ideal of D with Q ∩ R = (0).By [12, Proposition 3.10], the extension R ⊆ D is w -compatible if and only if D is t -linkedover R .We recall that, viewing ∗ as a (semi)star operation on R , an overring D is ∗ -compatibleover R if and only if D = D ∗ (see for example [16, Corollary 2.2]). This implies that a t -compatible extension is also w -compatible (i.e., t -linked).Flat extensions and generalized rings of fractions are examples of t -compatible extensions.In addition, the endomorphism rings of w , t or v -ideals of R are t -linked. Proposition 2.1.
Let R be a domain and let D be a t -compatible Mori overring of R . If R is Clifford t -regular (respectively, Boole t -regular, t -stable, strongly t -stable), then D isClifford t D -regular (respectively, Boole t D -regular, t D -stable, strongly t D -stable).Proof. If D is Mori each ideal of D is t D -finite, that is of type I t D with I finitely generated.Since R and D have the same field of fractions, I is an ideal of R . By [16, Lemma 2.4], itfollows that if I is Clifford t -regular (respectively, Boole t -regular, t -stable, strongly t -stable)as an ideal of R , then it is Clifford t D -regular (respectively, Boole t D -regular, t D -stable,strongly t D -stable) as an ideal of D . (cid:3) Corollary 2.2.
Let R be a Mori domain and let D be a generalized ring of fractions of R .If R is Clifford t -regular (respectively, Boole t -regular, t -stable, strongly t -stable), then D isClifford t D -regular (respectively, Boole t D -regular, t D -stable, strongly t D -stable).Proof. It follows from Proposition 2.1, because a generalized ring of fractions of a Moridomain is a t -compatible Mori domain. (cid:3) Remark 2.3. A t -linked overring of a Mori domain need not be Mori. In fact, if R is a Moridomain and each t -linked overring of R is Mori, then R has t -dimension one [9, Proposition2.20]. The converse holds for strong Mori domains [33, Theorem 3.4].However a t -compatible fractional overring of a Mori domain R is a Mori domain. In fact,by the next Proposition 3.1, any fractional overring D of R is ˙ t -Noetherian. If, in addition, D is t -compatible (i.e., D t = D ), ˙ t is a (semi)star operation on D . Hence ˙ t ≤ t D and so D isalso t D -Noetherian, that is Mori. We also observe that each t -compatible fractional overring D of a domain R is of type E ( I t ) := ( I t : I t ), for some ideal I of R . To see this, note that if D = D t is an ideal of R , we can write D = x − I t for some (integral) ideal I of R . Whence D = ( D : D ) = ( I t : I t ).We recall that for any domain R and ∗ = d, w, t , if R is Clifford ∗ -regular (respectively,Boole ∗ -regular, ∗ -stable, strongly ∗ -stable), each ring of fractions of R has the same property[16, Corollary 2.6(a)]. We now prove a converse for Mori domains.The following lemma is an easy calculation and follows from the fact that each ideal of aMori domain is t -finite. STEFANIA GABELLI AND GIAMPAOLO PICOZZA
Lemma 2.4.
Let R be a Mori domain and let R S be a ring of fractions of R . Then, denotingby t S the t -operation on R S , for any two ideals J and I of R , we have ( J t : I t ) R S = ( J t R S : I t R S ) and I t R S = ( IR S ) t S . Proposition 2.5.
Let R be a Mori domain. Then: (1) R is Clifford t -regular (respectively, t -stable, w -stable) if and only if R M is Clifford t M -regular (respectively, t M -stable, stable) for each M ∈ t -Max( R ) ; (2) If R is Clifford w -regular, then R M is Clifford regular, for each M ∈ t -Max( R ) . If,in addition, R is strong Mori, R is Clifford w -regular if and only if R M is Cliffordregular for each M ∈ t -Max( R ) .Proof. (1) Assume that R M is t M -stable for each M ∈ t -Max( R ). Then, if I is a nonzero idealof R , by applying Lemma 2.4, we get ( I ( I t : I )) t R M = ( I ( I t : I ) R M ) t M = ( IR M (( IR M ) t M :( IR M ) )) t M = (( IR M ) t M : ( IR M ) t M ) = ( I t : I t ) R M . Whence ( I ( I : I )) t = ( I t : I t ) and R is t -stable.In the same way, if R M is t M -regular, we get I t R M = ( IR M ) t M = (( IR M ) (( IR M ) t M :( IR M ) )) t M = (( I t ) R M ( I t R M : ( I ) t R M )) t M = ( I ( I t : I )) t R M , for each M ∈ t -Max( R )and so I t = ( I ( I t : I )) t .Conversely, if R is t -regular (respectively, t -stable), R M is t M -regular (respectively, t M -stable) by Corollary 2.2.The result for w -stability follows from [15, Corollary 1.10], since a Mori domain has the t -finite character.(2) If R is Clifford w -regular, R M has the same property by [16, Corollary 2.6](a). Butsince R is Mori, M R M is a t M -ideal (Lemma 2.4) and so the w -operation is the identity on R M .If R is strong Mori and R M is Clifford regular for each M ∈ t -Max( R ), to show that R isClifford w -regular we can proceed as in (1), recalling that if R is strong Mori each w -ideal is w -finite and so ( I w : I ) R M = ( IR M : I ), for each ideal I and M ∈ t -Max( R ). (cid:3) We now show that a Mori Clifford w -regular domain is t -stable, so that for Mori domains w -stable ⇒ w -regular ⇒ t -stable ⇒ t -regular.The following is an example of a Mori t -stable domain that is not ( w )-stable (this answersthe question in [15, Remark 1.7(1)]). We will see in Section 2 that a Noetherian Clifford t -regular domain of t -dimension strictly greater than one cannot be w -stable (equivalently w -regular). Example 2.6.
Let R be an integrally closed pseudo-valuation domain arising from a pullbackdiagram of type R −−−→ k y y V ϕ −−−→ K := VM where V := ( V, M ) is a DVR and R = V . Note that R is Mori one-dimensional [1, Theorem2.2], so that d = w and t = v on all the ideals of R .Since an integrally closed stable domain is Pr¨ufer [32, Proposition 2.1], then R is notstable. However R is (strongly) t -stable; in fact, for each divisorial non-principal ideal J v TAR STABILITY AND STAR REGULARITY FOR MORI DOMAINS 7 of R , we have that J v = J V is an ideal of V [19, Proposition 2.14] and so it is principal in V = ( J v : J v ). Lemma 2.7.
In a Mori finitely stable domain R every t -ideal is stable. In particular R is t -stable.Proof. Let I = J t , with J finitely generated, a t -ideal of R . Since R is finitely stable, J isinvertible in E ( J ) := ( J : J ). Since E ( J ) ⊆ E ( I ) := ( I : I ), J E ( I ) is invertible in E ( I ).Since ( J E ( I )) t = ( J t E ( I )) t = I , we get E ( I ) = J E ( I )( E ( I ) : J E ( I )) = J E ( I )( E ( I ) :( J E ( I )) t ) = J E ( I )( E ( I ) : I ) ⊆ I ( E ( I ) : I ) ⊆ E ( I ). It follows that I is invertible in E ( I ). (cid:3) Proposition 2.8.
Let R be a Mori domain. If R is Clifford w -regular, then every t -ideal of R is stable. In particular R is t -stable.Proof. In a Mori domain, the property that each t -ideal is stable is a t -local property.In fact, assume that I t R M is stable, for each M ∈ t -Max( R ). By using Lemma 2.4 wehave E ( I t ) R M = E ( I t R M ) = I t R M ( E ( I t R M ) : I t R M ) = I t ( E ( I t ) : I t ) R M , for each M ∈ t -Max( R ). So that E ( I t ) = I t ( E ( I t ) : I t ). Thus it is enough to show that every t M -ideal of R M is stable, for each M ∈ t -Max( R ). By Proposition 2.5(2), we know that R M is Clifford regular; hence R M is finitely stable [4, Proposition 2.3]. Since R M is Mori, weconclude by Lemma 2.7. (cid:3) Noetherian and strong Mori domains
A Clifford (indeed Boole) t -regular Noetherian domain need not be t -stable. In fact Kabbajand Mimouni gave an example of a Boole t -regular local Noetherian domain whose maximalideal is divisorial of height two [23, Example 2.4], while as we will see in a moment a t -stableNoetherian domain has t -dimension one. This last fact was proved in [23, Lemma 2.7] underthe stronger hypothesis that each t -ideal of R is stable.We work in the more general context of ∗ -Noetherian domains. Proposition 3.1.
Let R be a domain, ∗ a star operation on R , D a fractional overring of R . If R is ∗ -Noetherian, then D is ˙ ∗ -Noetherian.Proof. A ˙ ∗ -ideal of D is in particular a (fractional) ∗ -ideal of R . So the ascending chaincondition on ∗ -ideals of R implies the ascending chain condition on ˙ ∗ -ideals of D . (cid:3) Recall that, for a star operation ∗ on R , the ∗ -integral closure of R is the integrally closedoverring of R defined by R [ ∗ ] := S { ( J ∗ : J ∗ ); J ∈ F ( R ) finitely generated } [13]. When ∗ = d is the identity, we obtain the integral closure of R , here denoted by R ′ . If e R := S { ( I v : I v ); I ∈ F ( R ) } is the complete integral closure of R , we have R ⊆ R ′ ⊆ R [ ∗ ] ⊆ e R .As shown in [16, Theorem 4.1], when ∗ = e ∗ , each pair R, D with R ⊆ D ⊆ R [ ∗ ] satisfies astar version of Lying Over, Going Up and Incomparability. It follows that the ∗ -dimensionof R and the ˙ ∗ -dimension of R [ ∗ ] are the same [16, Corollary 4.2]. Proposition 3.2.
Let ∗ be a star operation on a domain R and assume that R is e ∗ -Noetherian. If each ∗ -maximal ideal M of R is a t E -invertible t E -ideal in E := E ( M ) ,then ∗ -dim( R ) = 1 = t -dim( R ) . STEFANIA GABELLI AND GIAMPAOLO PICOZZA
Proof.
We adapt the proof of [23, Lemma 2.7]. Assume that R has ∗ -dimension greater thanone. Note that ∗ is of finite type (since R is e ∗ -Noetherian and so also ∗ -Noetherian). Let P be a height-one prime ideal of R . Since P is a t -ideal, it is also a ∗ -ideal. So, there exists a ∗ -maximal ideal M of R containing P . Assume M = P and let E := E ( M ) := ( M : M ).Since M is a ∗ -ideal, it is also a e ∗ -ideal, so it is e ∗ -finite, i.e., M = J e ∗ for some finitelygenerated ideal J of R . Thus E = ( J e ∗ : J e ∗ ) ⊆ R [ e ∗ ] . So, by e ∗ -GU, there exist two ˙ e ∗ -ideals Q ( Q contracting respectively to P and M in R . Let Q be a prime ideal of E minimalover M in E such that Q ⊆ Q . We want to show that Q has height one. Note that E is˙ e ∗ -Noetherian (Proposition 3.1) and so also strong Mori. Assume that M is a t E -invertible t E -ideal of E . Since Q is minimal over M , Q is a t E -prime of E and so Q ⊆ N for some t E -maximal ideal N of E . Now, M is t E -invertible in E and so M E N = E N is principal.Since E N is Noetherian (because E is strong Mori), by the Principal Ideal Theorem, QE N has height one. It follows that Q has height-one.So Q ( Q and Q ∩ R = Q ∩ R = M , contradicting e ∗ -INC. It follows that R has ∗ -dimension one. Since ∗ ≤ t , we have ∗ -dim( R ) = 1 = t -dim( R ). (cid:3) Corollary 3.3.
Let R be a domain and ∗ a star operation on R . If R is e ∗ -Noetherian andall the ∗ -maximal ideals of R are ∗ -stable, then ∗ -dim( R ) = 1 = t -dim( R ) .In particular: (1) If R is Noetherian and ∗ -stable, then ∗ -dim( R ) = 1 = t -dim( R ) . (2) If R is strong Mori and t -stable, then t -dim( R ) = 1 .Proof. A ∗ -maximal ideal M of R is ˙ ∗ -invertible in E := E ( M ); thus it is t E -invertible(note that ˙ ∗ is of finite type since R is ∗ -Noetherian) and M = M ˙ ∗ = M t . Thus M is a t E -invertible t E -ideal of E ( M ) and we can apply Proposition 3.2.For (1), note that a Noetherian domain is e ∗ -Noetherian for every ∗ . (2) follows for ∗ = t . (cid:3) Corollary 3.4.
Let R be a e ∗ -Noetherian domain. If R is Clifford ∗ -regular and each ∗ -maximal ideal M of R is a t E -ideal of E := E ( M ) , then ∗ -dim( R ) = 1 = t -dim( R ) .Proof. As a consequence of Proposition 1.1(2), in a Clifford ∗ -regular e ∗ -Noetherian domainevery ∗ -maximal ideal M is t E -invertible in E . Hence we can apply Proposition 3.2. (cid:3) Remark 3.5.
Kabbaj and Mimouni showed that in a pullback diagram of type R −−−→ k y y T ϕ −−−→ K := TM where T := ( T, M ) is local Noetherian with maximal ideal M , R is a Boole t -regular domainif and only if so is T [23, Proposition 2.3].In this way it is possible to construct, as in [23, Example 2.4], examples of Boole t -regularlocal Noetherian domains of t -dimension greater than one. Namely, if T is a Noetherian UFDof dimension n ≥ K : k ] is finite, then R is a Boole t -regular local Noetherian domainwhose maximal ideal M is divisorial of height n . We note that since M is not divisorial in TAR STABILITY AND STAR REGULARITY FOR MORI DOMAINS 9 T , then T = ( M : M ) =: E ( M ) [14, Proposition 2.7] and so, according to Corollary 3.4, M is not a t E -ideal of E := E ( M ).Next, we prove a technical result that can be applied both to Noetherian ∗ -regular domainsand to strong Mori t -regular domains. Proposition 3.6.
Let ∗ = e ∗ ≤ ∗ be two star operations on a domain R . Assume that R is ∗ -Noetherian of ∗ -dimension one. The following conditions are equivalent for an ideal I of R : (i) I is Clifford ∗ -regular (respectively, Boole ∗ -regular) ; (ii) I is ∗ -stable (respectively, strongly ∗ -stable); (iii) I ∗ is ˙ ∗ -invertible (respectively, principal) in E ( I ∗ ) := ( I ∗ : I ∗ ) .Hence R is Clifford ∗ -regular (respectively, Boole ∗ -regular) if and only if R is ∗ -stable(respectively, strongly ∗ -stable).Proof. (i) ⇒ (iii) Assume that I is Clifford ∗ -regular. The overring E := E ( I ∗ ) := ( I ∗ : I ∗ ) is ˙ ∗ -Noetherian by Proposition 3.1. Moreover, E has ˙ ∗ -dimension one. Indeed, I ∗ isa ∗ -ideal, so by ∗ -Noetherianity there exists a finitely generated ideal H of R such that I ∗ = H ∗ . Hence E = E ( H ∗ ) ⊆ R [ ∗ ] and the ˙ ∗ -dimension of E is one by [16, Corollary4.2]. Since ∗ is of finite type, ˙ ∗ ≤ t E and E has t E -dimension one. Hence t E -Max( E ) = ˙ ∗ -Max( E ) and so t E -invertibility coincides with ˙ ∗ -invertibility. Now I ∗ = H ∗ = H ∗ is t E -invertible in E by Proposition 1.1(2). Thus I ∗ is ˙ ∗ -invertible.If I is Boole ∗ -regular, we have just seen that I ∗ is ˙ ∗ -invertible in E . Since ∗ ≤ ∗ , I is also ˙ ∗ -invertible. Hence I is strongly ∗ -stable by Proposition 1.1(3).(iii) ⇒ (ii) is clear since ∗ ≤ ∗ and (ii) ⇒ (i) by Proposition 1.1(1). (cid:3) Corollary 3.7.
Let R be a one-dimensional Noetherian domain and ∗ a star operation on R . The following conditions are equivalent for an ideal I of R : (i) I is Clifford ∗ -regular (respectively, Boole ∗ -regular); (ii) I is ∗ -stable (respectively, strongly ∗ -stable); (iii) I ∗ is invertible (respectively, principal) in E ( I ∗ ) .Hence R is Clifford ∗ -regular (respectively, Boole ∗ -regular) if and only if R is ∗ -stable(respectively, strongly ∗ -stable). In addition, under any of these conditions e ∗ = d .Proof. Apply Proposition 3.6 for ∗ = d .In addition, if R is ∗ -stable, e ∗ = w [15, Corollary 1.6] and if R is one-dimensional w = d . (cid:3) In the local one-dimensional Noetherian case ∗ -stability and Boole ∗ -regularity are equiv-alent. Proposition 3.8.
Let R be a local one-dimensional Noetherian domain and ∗ a star opera-tion on R . The following conditions are equivalent for an ideal I of R : (i) I is Clifford ∗ -regular; (ii) I is ∗ -stable; (iii) I ∗ is invertible in E ( I ∗ ) ; (iv) I ∗ is principal in E ( I ∗ ) ; (v) I is Boole ∗ -regular.Thus R is Clifford ∗ -regular if and only if R is Boole ∗ -regular if and only if R is (strongly) ∗ -stable. In addition, under any of these conditions e ∗ = d .Proof. (i) ⇔ (ii) ⇔ (iii) and (iv) ⇔ (v) by Corollary 3.7. (v) ⇒ (i) is clear.(iii) ⇒ (iv) Since R is Noetherian, E := E ( I ∗ ) is a finitely generated R -algebra and E isintegral over R . By [7, Chapitre 5, §
2, n. 1, Proposition 3] the number of prime ideals of E contracting to the maximal ideal of R is finite. Hence E is semilocal and it follows that I ∗ is principal in E .Finally e ∗ = d by Corollary 3.7. (cid:3) Remark 3.9.
Proposition 3.8 does not hold in the non-local case. In fact, any Dedekinddomain that is not a PID furnishes an example of a one-dimensional Noetherian domain thatis stable but not strongly stable.For Boole t -regularity, the equivalence of conditions (i) and (ii) in the next corollary wasproven in [23, Theorem 2.10]. Corollary 3.10.
Let R be a strong Mori domain. The following statements are equivalent: (i) R is Clifford t -regular (respectively, Boole t -regular) of t -dimension one; (ii) R is t -stable (respectively, strongly t -stable); (iii) R is Clifford t -regular (respectively, Boole t -regular) and the t -maximal ideals of R are t -stable.Proof. (i) ⇔ (ii) follows by Proposition 3.6 (for ∗ = w and ∗ = t ) and the fact that a t -stable strong Mori domain has t -dimension one (Corollary 3.3).(ii) ⇒ (iii) is obvious and (iii) ⇒ (i) follows by Corollary 3.3. (cid:3) By [16, Proposition 1.6], w -regularity and w -stability coincide on strong Mori domains,because each w -ideal is w -finite. We now give another proof by using Proposition 3.6. Corollary 3.11.
Let R be a strong Mori domain. The following statements are equivalent: (i) R is Clifford w -regular (respectively, Boole w -regular); (ii) R is w -stable (respectively, strongly w -stable).Proof. A Clifford w -regular strong Mori domain is t -stable (Proposition 2.8) and so it has t -dimension one (Corollary 3.3). Hence we can apply Proposition 3.6 for ∗ = ∗ = w . (cid:3) Motivated by the fact that a Krull Boole t -regular domain is a UFD [21, Proposition2.2], Kabbaj and Mimouni ask whether the integral closure of a Noetherian Boole t -regulardomain is a UFD [23, Question 2.11(3)]. The answer is positive for w -regularity (recall thatin a Krull domain t = w ). Proposition 3.12.
Let R be a strong Mori domain. If R is Boole w -regular then R [ w ] is aUFD.Proof. If R is strong Mori then R [ w ] is a Krull domain [8, Theorem 3.1]. In addition, if R isBoole w -regular, then R [ w ] is a GCD-domain [16, Theorem 4.3]. Thus R [ w ] is a UFD. (cid:3) TAR STABILITY AND STAR REGULARITY FOR MORI DOMAINS 11
Corollary 3.13.
Let R be a Noetherian domain. If R is Boole w -regular (respectively,regular) then R ′ is a UFD (respectively, a PID).Proof. Just recall that, when R is Noetherian, R ′ = R [ w ] and that Clifford regularity implies d = w . (cid:3) Mori domains
The proof of Proposition 3.2 cannot be extended to Mori domains, since it is based onthe Principal Ideal Theorem. However we now show that ( w -)stable Mori domains have( t -)dimension one (cf. [20, Lemma 4.8]). Proposition 4.1.
A stable (respectively, w -stable) Mori domain has dimension (respectively, t -dimension) one.Proof. By a corrected version of [28, Corollary 2.7], a local domain R is stable if and only ifone of the following conditions holds:(a) R is one-dimensional stable;(b) R is a strongly discrete valuation domain;(c) R arises from a pullback diagram of type: R −−−→ D y y V ϕ −−−→ VI where V is a strongly discrete valuation domain, I is an ideal of V , D is a local stable ringof dimension at most one having a prime ideal P such that P contains all the zero-divisorsof D and P = (0), and V /I is isomorphic to the total quotient ring of D [30].Since a Mori valuation domain is a DV R and in a diagram as in (c) R is Mori if and onlyif V is a DV R and D is a field [25, Theorem 9], we see that a local Mori stable domain hasdimension one.To conclude, recall that a ( w -)stable domain is ( t -)locally stable [15, Corollary 1.10]. (cid:3) When ∗ is the identity, the analog of Proposition 3.8 was proved for Mori domains in [16]. Proposition 4.2. [16, Corollary 3.4]
Let R be a local one-dimensional Mori domain. Thefollowing conditions are equivalent: (i) R is Clifford regular; (ii) R is (strongly) stable; (iii) R is Boole regular. In [16, Theorem 4.8] we showed that in t -dimension one w -regularity and w -stability areequivalent on a domain R if and only if the w -integral closure R [ w ] is a Krull domain. ForMori domains we have the following result. Proposition 4.3.
Let R be a Mori domain. The following conditions are equivalent: (i) R is Clifford w -regular of t -dimension one (respectively, regular of dimension one); (ii) R is Clifford w -regular and R [ w ] is a Krull domain (respectively, regular and R ′ is aDedekind domain); (iii) R is w -stable (respectively, stable).Under (anyone of ) these conditions R [ w ] = e R (respectively, R ′ = e R ) is the complete integralclosure of R .Proof. Since Clifford regularity and stability imply d = w , it is enough to prove the theoremfor the w -operation.(i) ⇒ (iii) For each M ∈ t -Max( R ), R M is one-dimensional Mori and Clifford regularby Proposition 2.5(2). Hence R M is stable by Proposition 4.2. We conclude by applyingProposition 2.5(1).(iii) ⇒ (ii) If R is w -stable, D := R [ w ] is a w | D -stable overring of R [15, Corollary 2.6],hence D is a P v MD with t -finite character such that ( P ) t = P , for all P ∈ t -Spec( R ) [15,Theorem 2.9]. Since R has w -dimension one (Proposition 4.1), D has ˙ w -dimension one (seefor example [16, Corollary 4.2]) and since ˙ w ≤ t D it has also t D -dimension one. It followsthat D := R [ w ] is a Krull domain.(ii) ⇒ (i) If D = R [ w ] is Krull, it has w D -dimension one. Now R is Clifford w -regular, sothat ˙ w = w D [16, Theorem 4.3]. Hence we conclude that R has w -dimension one (and so, t -dimension one) by [16, Corollary 4.2].Since a Krull domain is completely integrally closed, if R [ w ] is Krull, we have g R [ w ] = R [ w ] .Hence, from R ⊆ R [ w ] ⊆ e R , we obtain e R ⊆ g R [ w ] = R [ w ] ⊆ e R and R [ w ] = e R . (cid:3) If R is strong Mori then R [ w ] is a Krull domain [8, Theorem 3.1], hence from Theorem 4.3we obtain again that w -stability and w -regularity coincide on strong Mori domains, as seenin Corollary 3.11.Proposition 4.3 improves [20, Theorem 4.7]. In relation to this result, Kabbaj and Mimouniask whether a local Mori Clifford regular domain is one-dimensional if and only if its completeintegral closure e R is Dedekind [20, page 633]. We can give the following answer. Corollary 4.4.
Let R be a Mori Clifford w -regular (respectively, regular) domain. Thefollowing conditions are equivalent: (i) R has t -dimension one (respectively, dimension one); (ii) R [ w ] is a Krull domain (respectively, R ′ is a Dedekind domain).Under (anyone of ) these conditions, R is w -stable and R [ w ] = e R (respectively, R ′ = e R ) isthe complete integral closure of R . Proposition 4.5.
Let R be a Mori integrally closed domain. Then the following conditionsare equivalent: (i) R is Clifford w -regular; (ii) R is a Krull domain; (iii) R is w -stable.Proof. (i) ⇒ (ii) Since an integrally closed Clifford w -regular domain is a P v MD [16, Corol-lary 4.5], R is Krull.(ii) ⇒ (iii) because in a Krull domain each t -ideal is t -invertible and t = w . TAR STABILITY AND STAR REGULARITY FOR MORI DOMAINS 13 (iii) ⇒ (i) is clear. (cid:3) If R is Mori and w -divisorial , that is w = t = v (as star operations), R is a strong Moridomain of t -dimension one [11, Corollary 4.3]. If, in addition, R is also w -stable, each domain D t -linked over R is w D -divisorial [15, Corollary 3.6]. Hence from Corollaries 3.10 and 3.11,we obtain the following result. Proposition 4.6.
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