INTEGRAL DOMAINS IN WHICH NONZERO LOCALLY PRINCIPAL IDEALS ARE INVERTIBLE

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1 INTEGRAL DOMAINS IN WHICH NONZERO LOCALLY PRINCIPAL IDEALS ARE INVERTIBLE D.D. ANDERSON AND MUHAMMAD ZAFRULLAH A. Westudylocallyprincipalidealsandintegraldomains,calledLPI domains, in which every nonzero locally principal ideal is invertible. We show that a finite character intersection of LPI overrings is an LPI domain. Hence ifadomaindisafinitecharacterintersectiond= D P forsomesetofprime idealsofd,then Disan LPIdomain. Bazzoniin[10]andin[11]putforwardtheconjecture: IfDisaPrüferdomain suchthateverynonzerolocallyprincipalidealofdisinvertible,thendisoffinite character. (A domain D is Prüfer if every nonzero finitely generated ideal of D is invertible and D is of finite character if every nonzero nonunit of D belongs to only finitely many maximal ideals of D.) This conjecture was resolved in the affirmative by Holland, Martinez, McGovern, and Tesemma in[18]. Later Halter- Koch[16] stated and proved an analog of Bazzoni s conjecture for r-prüfer monoids, which in the domain case are PVMD s(defined below) and include Prüfer domains. Recently, in[23], the second author has treated the Bazzoni Conjecture in a simpler manner encompassing the results of the above mentioned authors. This note is to record the results proved in an effort to answer the following question. What are the domains, called LPI domains, that have the property LPI: every nonzero locally principal ideal is invertible? Our main result is that a finite character intersection of LPI overrings is an LPI domain. Hence if D has a set S of prime ideals with D = P S D P being of finite character, D is an LPI domain. As our work will involve the use of star operations, we provide below a quick review. Mostoftheinformation givenbelowcanbefound in[22] and[13, sections32, 34], alsosee[15]. LetD denote anintegraldomain withquotientfield K andlet F(D)be the setofnonzero fractional ideals of D. Astar operation on D is a function :F(D) F(D)suchthatforallA,B F(D)andforall0 x K (a)(x) =(x)and(xa) =xa, (b)a A anda B whenevera B,and (c)(a ) =A. ForA,B F(D)wedefine -multiplication by(ab) =(A B) =(A B ). A fractionalideala F(D)iscalleda -ideal ifa=a anda -idealaisoffinite type if A=B where B is afinitely generated fractional ideal. Astar operation is said to be of finite character if A = {B 0 B is a finitely generated subidealofa}. ForA F(D)defineA 1 ={x K xa D}andcallA F(D) -invertible if(aa 1 ) =D. Clearlyeveryinvertibleidealis -invertibleforevery staroperation.if isoffinitecharacterandais -invertible,thena isoffinite Date: June 2, Mathematics Subject Classification. Primary 13A15; Secondary 13F05; 13E99. Key words and phrases. Locally principal ideal, invertible ideal, finite t-character. 1

2 2 D.D. ANDERSON AND MUHAMMAD ZAFRULLAH type. The best known examples of star operations are the d-operation defined by A A d =A,thev-operation definedbya A v =(A 1 ) 1,andthet-operation definedbya A t = {B v 0 B isafinitelygeneratedsubidealofa}. Given twostaroperations 1, 2 ondwesaythat 1 2 ifa 1 A 2 foralla F(D). Notethat 1 2 ifandonlyif(a 1 ) 2 =A 2 foralla F(D), orequivalently, (A 2 ) 1 = A 2 for all A F(D). By definition t is of finite character, t v, and ρ t for every star operation ρ of finite character. If is a star operation of finite character, then using Zorn s Lemma we can show that a proper integral idealmaximalwithrespecttobeinga -idealisaprimeidealandthateveryproper integral -ideal is contained in a maximal -ideal. Let us denote the set of all maximal -ideals by -max(d). It can also be easily established that for a star operation of finite character on D, we have D = {D M M -max(d)}. A v-idealaoffinitetypeist-invertibleifandonlyifaist-locallyprincipal,i.e.,for every M t-max(d) we have AD M principal. An integral domain D is called a Prüfer v-multiplication domain (PVMD) if every nonzero finitely generated ideal ofdist-invertible. AccordingtoGriffin[14,Theorem5]DisaPVMDifandonly ifd M isavaluationdomainforeachm t-max(d).adomaindissaidtobeof finite character (resp., finite t-character) if every nonzero nonunit of D belongs to only a finite number of maximal ideals(resp., maximal t-ideals), or equivalently, the intersectiond= {D M M max(d)}(resp.,d= {D M M t-max(d)})isof finitecharacterorislocallyfinite,i.e.,eachnonzeroelementofd(ork)isaunitin almostalld M. Moregenerally,wesaythatDisoffiniteprimecharacter (orfinite S-character,ifweneedtomentionthesetS)ifthereexistsasetSofprimeideals of D with D = P S D P locally finite. We can now state the PVMD analog of Bazzoni s conjecture which was proved by Halter-Koch[16] and the second author [23]: apvmddisoffinitet-characterifandonlyifeveryt-locallyprincipalt-ideal of D is t-invertible. RecallthatanidealI in aringriscalled a cancellation ideal ifij =IK for idealsjandkofrimpliesthatj=k. Notethatinthisdefinitionwecanreplace = by. In[7]itwasshownthatanidealI isacancellationidealifandonly ifforeachmaximalidealm ofr,i M isaregularprincipalidealofr M. Withthis in mind, we have the following characterization of nonzero locally principal ideals in an integral domain. Theorem 1. For an integral domain D and nonzero ideal I of D, the following conditions are equivalent. (1) I is locally principal. (2) I isacancellationideal. (3) I isfaithfullyflatasad-module. Proof. (1) (2)Thisisgivenin[7,Theorem]asmentionedinthepreviousparagraph. (1) (3) This follows since being faithfully flat is a local condition. (3) (2)SupposethatIJ IKforidealsJandKofD. ThenI D ((J+K)/K)= I D (J+K)/I D K =I(J+K)/IK =IK/IK =0; so(j+k)/k =0, that is, J K. Herethefirsttwoequalitiesfollowfromthe flatnessofi andthelast equality from the faithfulness. With this preparation we return to the question: What integral domains satisfy the property LPI: every nonzero locally principal ideal is invertible? Now it is well known thatanonzeroideal is invertible ifand only ifit is finitely generated and

3 LOCALLY PRINCIPAL IDEALS 3 locally principal[20, Theorems 58 and 62]. So this gives the criterion that a nonzero locally principal ideal is invertible if(and only if) it is finitely generated. Thus a Noetherian domain is an LPI domain. Also, according to Lemma 37.3 of Gilmer [13], stated below for integral domains (with our addition of the last statement), every semi-quasi-local domain is an LPI domain. In fact, in a semi-quasi-local domain a locally principal ideal is actually principal [20, Theorem 60]. However, the result that we would really like, namely that a locally principal ideal contained in only finitely many maximal ideals is invertible, is not true. For [13, Example 42.6]givesanexampleofanalmostDedekinddomainD(i.e.,DislocallyaDVR) withexactlyonenoninvertiblemaximalidealm. ThusM iscontainedinaunique maximal ideal and is locally principal, but M is not invertible. Proposition1. ([13,Lemma37.3])Letx Dsuchthatxbelongstofinitelymany maximal ideals M 1,M 2,...,M n ofd. If Ais an ideal of D containing x suchthat AD Mi isfinitelygeneratedforeachibetween1andn,thenaisfinitelygenerated. (Hence if A is nonzero locally principal, A is invertible.) We next give a slight extension of Proposition 1 which has the added benefit that its converse is also true. Theorem 2. LetDbeanintegral domain andi anonzerolocallyprincipal ideal (resp., locally finitely generated ideal) of D that is contained in only finitely many maximal ideals. Then I is invertible (resp. finitely generated) if and only if there exitsafinitelygeneratedideala I suchthataiscontainedinonlyfinitelymany maximal ideals. In the invertible case I can be generated by two elements. Proof. ( ) If I is finitely generated(which includes the invertible case), we can take AtobeI. ( )SupposethatAiscontainedinthemaximalidealsM 1,M 2,...,M n. If I M i for some i, then for x I M i, we can replace A by (A,x) and (A,x) M i. ThuswecanassumethatM 1,M 2,...,M n arealsothemaximalideals containingi. Foreachi,chooseafiniteset{b ij }iniwithd Mi (b ij )=I Mi. Replace A by (A,{b 1j },...,{b nj }). Then A Mi =I Mi for each i and A N = D N = I N for eachothermaximalidealn ofd. SoA=I locallyandhenceglobally. ThusI is finitely generated. Hence if I is locally principal, then I is finitely generated and locally principal and thus invertible. The last statement follows from[13, Exercise 9, Section 7]. What is really behind the fact that a nonzero finitely generated locally principal idealisinvertibleisthefactthatforidealsi andj ofacommutativeringrwith I finitelygeneratedandanymultiplicativelyclosedsets ofr,wehave(j:i) S = (J S :I S ). Wenextgiveacharacterizationofinvertibleidealsusingthis. Thelast statementofthenexttheoremisaspecialcaseof[4,theorem12]. Theorem 3. (a)foranonzerolocallyprincipal ideal I ofanintegraldomaind, the following conditions are equivalent. (1) I is invertible. (2) I is finitely generated. (3) Forallnonzerox I,(Dx: D I) M =(D M x: DM I M )foreachmaximal idealm ofd.

4 4 D.D. ANDERSON AND MUHAMMAD ZAFRULLAH (4) For some nonzero x I, (Dx : D I) M = (D M x : DM I M ) for each maximalidealm ofd. (5) (I 1 ) M =(I M ) 1 foreachmaximalidealm ofd. (b) LetI beanonzerolocallyprincipalidealofanintegraldomaind. Suppose thatforsomenonzerox I,Dxhasaprimarydecomposition. ThenI is invertible. Thus an integral domain in which every proper principal ideal has a primary decomposition is an LPI domain. Proof. (a) (1) (2) (3) (4) Clear. (4) (1) Let M be a maximal ideal ofd. Now((Dx:I)I) M =(D M x:i M )I M =D M xwherethelastequalityholds since I M is principal. Thus (Dx : I)I = Dx locally and hence globally. Since I isafactorofanonzeroprincipalideal,i isinvertible. (2) (5)(I 1 ) M =[D: K I] M =[D M : K I M ]=(I M ) 1 wherethesecondequalityfollowssincei isfinitely generated. (5) (1)(II 1 ) M =I M (I 1 ) M =I M (I M ) 1 =D M foreachmaximal idealm ofd. HerethelastequalityholdsbecauseI M isprincipal. SoII 1 =D andhencei isinvertible. (b)supposethatdx=q 1 Q n whereeachq i is primary. Then(Dx:I)=(Q 1 Q n :I)=(Q 1 :I) (Q n :I). Nowforany multiplicativelyclosedsetsandanyprimaryidealqwehave(q:i) S =(Q S :I S ). (See,forexample,[4,Lemma11].) Hence(Dx:I) S =(Q 1 Q n :I) S =((Q 1 : I) (Q n :I)) S =(Q 1 :I) S (Q n :I) S =(Q 1S:I S ) (Q ns :I S )= (Q 1S Q ns :I S )=(D S x:i S ). By(a)I isinvertible. While the condition that the nonzero locally principal ideal I contains a nonzero principal ideal having a primary decomposition is sufficient for I to be invertible, it is by no means necessary. For example, if V is a two-dimensional valuation domain and x is a nonzero element of V contained in a height-one prime ideal, thencertainlyvxisinvertible,butvxcontainsnononzeroprincipalidealhaving a primary decomposition. Ofcourse,usingProposition1orTheorem2,weconcludethatifDisoffinite character, then nonzero locally principal ideals are invertible and can be generated bytwoelements. NotethattheexampleofaDedekinddomainthatisnotaPID showsthatwecannotreplace twoelements by oneelement hereorintheorem 2. Recallthatif{D α }isafamilyofoverringsofd(ringsbetweendanditsquotient field K) such that D = D α, then the function A AD α is a star operation which is offinitecharacter if D= D α isoffinite character[2, (4)Theorem2]. Also,observethatifIisanonzerolocallyprincipalidealofDandT isanoverring ofd(ormoregenerallyanyintegraldomaincontainingdasasubring),thenit is (nonzero) locally principal. If IT =T, this is clear. Suppose that IT M, a maximalidealoft,andletp =M D. ThenID P isprincipal(beingalocalization ofid N foranymaximalidealn ofdcontainingp). Hence(IT) M =ID P T M is principal. Thus by Theorem 1, IT is also a cancellation ideal. Wehaveonemoreresulttoquotebeforewegivethemainresultofthisnote. Lemma 1. ([1, Theorem 2.1]) Let I be a nonzero locally principal ideal in an integraldomain. ThenI is at-ideal. Further, I isinvertibleifandonlyifi is of finite type.

5 LOCALLY PRINCIPAL IDEALS 5 Note that while a nonzero locally principal ideal is a t-ideal, it need not be a v-ideal. (Ofcourse,aninvertibleidealisav-ideal.) Forexample,ifDisanalmost Dedekind domain, then a maximal ideal M of D is locally principal, but M is a v-idealifandonlyifm isinvertible. Notethatalocallyprincipalv-idealneednot beinvertible. ForletDbeanintegraldomainthatislocallyaGCDdomain,but isnotaggcddomain. (RecallthatDisaGGCD domain iftheintersectionof two invertible(or equivalently, principal) ideals is invertible.) So there are nonzero x,y D with(x) (y)notinvertible. But(x) (y)isalocallyprincipalv-ideal, necessarilynotoffinitetype. Foranexampleofsuchadomain,see[17]. Withthis preparation we have our main result. Theorem 4. Let D be an integral domain where D = D α is a finite character intersectionofoverringsd α eachsatisfyinglpi.thendisanlpidomain. Hence if D has finite prime character (i.e., D has a set S of prime ideals such that D= P S D P isoffinitecharacter),thendisanlpidomain. Proof. Let bethestaroperationgivenbya AD α. Sincetheintersectionis of finite character, has finite character. Let I be a nonzero locally principal ideal ofd. Choose0 x I. Letα 1,...,α n betheindiceswithxd α D α. NowID αi is nonzero locally principal and hence invertible and thus finitely generated. So we canenlargedxtoafinitelygeneratedideala I suchthatid α =AD α foreach α. ThusI = ID α = AD α =A. Since hasfinitecharacterandi isat-ideal, we have I = I t = (I ) t = (A ) t = A t. So I has finite type. By Lemma 1, I is invertible. The second statement is now immediate because a quasi-local domain isanlpidomain. RecallthatanintegraldomainDisaMoridomain ifitsatisfiestheascending chain condition on divisorial ideals. So Mori domains include Noetherian domains andkrulldomains. Thereadermaysee[9]foranintroductiontotheseringsand for a recentbibliography. For ourpurposes let us note that in a Mori domain D everymaximalt-idealisdivisorialandfromtheorem3.3of[9]weconcludethata Mori domain is of finite t-character. Corollary 1. In any integral domain of finite character or finite t-character, every nonzero locally principal ideal is invertible. Consequently, a Mori domain is an LPI domain. WenextmentionaresultrelatedtothesecondstatementofTheorem4. Note thatthispropositioncanbeusedtogiveanalternateproofofthesecondstatement oftheorem4. ForifweassumethattheidealAinProposition2isnonzerolocally principal,thenabeingt-invertibleisoffinitetype. SobyLemma1,Aisinvertible. Proposition2. ([6,Lemma2.2])LetS beacollectionofnonzeroprimeidealsof anintegraldomaind. SupposethatD= P S D P wheretheintersectionhasfinite character. Let be the star operation A A = P S AD P. If A F(D) such thatad P isprincipalforeachp S,thenAis -invertibleandhencet-invertible. We next consider the question of when the polynomial ring D[X] is an LPI domain. WeshowthatifD[X]isanLPIdomain,thensoisDandforDintegrally closed, theconverseistrue. WhilewedonotknowingeneralwhetherDanLPI

6 6 D.D. ANDERSON AND MUHAMMAD ZAFRULLAH domainimpliesd[x]isanlpidomain,wenotethatdhasfiniteprimecharacter (resp., finite t-character) if and only if D[X] does. Theorem5. LetDbeanintegraldomain. (1) IfD[X] is anlpi domain,thensoisd. IfDis an integrallyclosedlpi domain,thend[x]isanlpidomain. (2) DhasfiniteprimecharacterifandonlyifD[X]hasfiniteprimecharacter. (3) D hasfinitet-characterifandonlyifd[x]hasfinitet-character. Proof. (1)SupposethatD[X]isanLPIdomain. LetAbeanonzerolocallyprincipalidealofD. ThenbytheremarksoftheparagraphprecedingLemma1,AD[X] isanonzerolocallyprincipalidealofd[x].sinced[x]isanlpidomain,ad[x] is invertible. Hence A is invertible. So D is an LPI domain. Conversely, suppose that D is an integrally closed LPI domain. Let B be a nonzero locally principal idealofd[x]. SinceDisintegrallyclosedandBisat-ideal,B= f g AD[X]where f,g D[X]andAisat-idealofD(see,forexample,[3,Corollary3.1]). NowB nonzero locally principal implies AD[X] is nonzero locally principal and hence A is nonzerolocallyprincipal. (ForletM beamaximalidealofd. ThenAD[X] (M,X) isprincipalandthusad[x] M[X] =AD M (X)isprincipal. ButthenAD M isprincipal.) SinceDis an LPI domain, Ais invertible. Thus AD[X] is invertible and hencesoisb. SoD[X]isanLPIdomain. (2) ( ) Suppose that D[X] has finite S-character: D[X] = Q S D[X] Q, locally finite. Let S = {Q D Q S}. Then D = P S D P is a finite S - character representation for D. For certainly this representation is locally finite. AndD= P S D P sinced=d[x] K =( Q S D[X] Q ) K = Q S (D[X] Q K) P S D P D since D[X] Q K D P where P = Q D. ( ) Suppose that D has finite S-character, so D = P S D P is locally finite. Now D[X] = P S D P [X](butisnotlocallyfinite)andD P [X]=D[X] P[X] K[X]. SoD[X]= ( P S D[X] P[X] ) ( Q T D[X] Q )isafiniteprimecharacterrepresentationwhere T = {Q Spec(D[X]) Q D = 0}. The locally finiteness follows since for f g K(X), the contents c(f) and c(g) of f and g, respectively, are contained inonlyfinitelymanyp;so f g isaunitinalmostalld[x] P[X]. (3)ThisisgivenforDintegrallyclosedin[21,Proposition4.2];butthehypothesisthatDisintegrallyclosedisnotused. Weofferasimplerproof. Recallfrom [19,Proposition1.1]thatifM isamaximalt-idealofd[x]withm D 0,then M = (M D)[X]. Noting that if A is an integral t-ideal of D, then A[X] is an integralt-idealofd[x],weconcludethataisamaximalt-idealofdifandonlyif A[X]isamaximalt-idealofD[X]. NowsupposethatD[X]isoffinitet-character andtakeanonzerononunitd D. Thendbelongstoonlyafinitenumberofmaximal t-ideals of D[X], each necessarily of the form M[X] where M is a maximal t-idealofd. SoDisoffinitet-character. Conversely,supposethatDisoffinite t-character. Letf beanonzerononunitofd[x]. Thenf belongstotwokindsof maximalt-idealsm: (a)m suchthatm D 0and(b)M suchthatm D=0. Nowtheonesin(a)arefiniteinnumberbecauseDisoffinitet-characterandthe onesin(b)beingupperstozeroarealsofiniteinnumber. We opened this paper by noting that a Prüfer domain D is an LPI domain if and only if D has finite character (or equivalently, finite t-character, since every

7 LOCALLY PRINCIPAL IDEALS 7 nonzero ideal of a Prüfer domain is a t-ideal). Thus examples of non-lpi domains include the ring of all algebraic integers and the ring of entire functions. Also, an almostdedekinddomainisanlpidomainifandonlyifitisdedekind. LetDbeanintegraldomainsuchthatforeachn 1everyproperprincipalideal ofd[x 1,...,X n ]hasaprimarydecomposition(e.g.,disnoetherian). Thenforany setofindeterminates{x α },everyproperprincipalidealofd[{x α }]hasaprimary decompositionandhenced[{x α }]isanlpidomain. In particular,z[{x α }]isan LPIdomain. However,everyringisahomomorphicimageofZ[{X α }]forsomeset ofindeterminates{x α }. ThusthehomomorphicimageofanLPIdomainneednot beanlpidomain. LetDbeanintegraldomainandletX (1) (D)bethesetofheight-oneprimeideals ofd. ThenDissaidtobeweaklyKrullifD= P X (1) (D)D P wheretheintersection is locally finite. Note that D is weakly Krull if and only if every nonzero proper principal ideal of D has a reduced primary decomposition involving only height-one primes[5, Theorem 13]. Thus for a one-dimensional integral domain D the following are equivalent: (1) D has finite character, (2) D has finite t-character, (3) D is weakly Krull, (4) every proper principal ideal of D has a primary decomposition, and(5)dislaskerian,i.e.,everyproperidealofdhasaprimarydecomposition. Note that D can be of finite prime character without each proper principal ideal having a primary decomposition. For example, a valuation domain V is of finite character and finite t-character, but each proper principal ideal of V has a primary decompositionifandonlyifdimv =1. However,noexamplecomestomindofa domain D in which every proper principal ideal has a primary decomposition, but Disnotoffiniteprimecharacter. Also,wehavenoexampleofanLPIdomainthat is not of finite prime character. Wenextgiveanexampleofaringoffinitecharacter(andhenceanLPIdomain) that is not of finite t-character. Thus an LPI domain need not be of finite t- character. Note that conversely, if D has finite t-character, then D need not have finitecharacterasseenbyd=k[x 1,...,X n ]wherek isafieldandn 2. Thus K[X 1 ] hasfinitecharacter, butk[x 1 ][X 2 ] does not; sointheorem5we can not add(4)disoffinitecharacterifandonlyifd[x]is. Example1. LetRbearegularlocalringofdimensiongreaterthan1withquotient fieldk andletx beanindeterminateoverk.thend=r+xk[x]isadomain that is of finite character and hence every locally principal nonzero ideal is invertible, butnotoffinitet-character. Infact,DcontainsanonzeroidealAthatist-locally principal, but not t-invertible. Illustration: ByTheorem4.21of[12]everymaximalidealofDiseitherofthe formm+xk[x]wheremisthemaximalidealofrorisoftheformf(x)dwhere f(0)=1. NowatypicalelementofD=R+XK[X]isoftheform a b Xr (1+Xf(X)) where b a if r = 0. If r = 0 and c = a b R, then in c(1+xf(x)), c is clearlycomaximalwith1+xf(x),c M+XK[X],and1+Xf(X)isaproduct of principal (maximal) primes. We conclude that c(1 + Xf(X)) belongs to only a finite number of maximal ideals. Next if r > 0 we note that as X does not belong to any prime ideal of the form (1+Xg(X))D, a b Xr does not belong to anymaximalidealscontaining1+xf(x),whichmeansthat1+xf(x)and a b Xr are comaximal. Noting that a b Xr M +XK[X], which is unique in this case, we conclude that a b Xr (1+Xf(X)) belongs to only a finite number of maximal ideals. HavingexhaustedallthecasesweconcludethatDisoffinitecharacterand

8 8 D.D. ANDERSON AND MUHAMMAD ZAFRULLAH consequently every nonzero locally principal ideal of D is invertible. To see that D isnotoffinitet-characterrecallfrompage437of[12]forp anonzeroprimeofr, P+XK[X]isamaximalt-idealofDifandonlyifP isamaximalt-idealofr.as R is a UFD with infinitely many non-associate principal primes we have infinitely many distinct maximal t-ideals of the form pr + XK[X] containing the element X. ToconstructtheidealAselectaset{p 1,p 2,...}ofnon-associateprincipalprimes of R and as in [23] we can construct A = ({p 1 1 p 1 n X} n=1) which is t-locally principal, yet not of finite type and hence not t-invertible. Weendbyconsideringthequestion: IfDisanLPIdomainandSisamultiplicativesetofD,mustD S beanlpidomain? Notethatifeveryproperprincipalideal ofdhasaprimarydecomposition,thenthesameistrueofd S. Note,however,that ifdhasfinitet-character,thend S neednotagainbeoffinitet-character. In[8, Example2c]isanexampleofadomainDoffinitet-characterwithamaximalideal M suchthatd M doesnothavefinitet-character. Butasintheabovementioned example the result is still a quasi-local ring we do not have a definite counterexampletothequestion. However,ifDisoffinitet-characterandt-Spec(D)istreed thentheanswerisyesforeverymultiplicativesets. Thuswehaveasimplespecial case. Moreover,wedonotknowofanexampleofanintegraldomainoffiniteprime charactersuchthatsomelocalizationd S doesnothavefiniteprimecharacterand we know of no example of an LPI domain with a localization that is not an LPI domain Thus we end with the following questions. Question1. IfDisanLPIdomain,isDoffiniteprimecharacter? Question2. IfDisanLPIdomain,isD[X]anLPIdomain? Question3. IfDisanLPIdomainandSisamultiplicativelyclosedsubsetofD, isd S anlpidomain? Question 4. If every proper principal ideal of D has a primary decomposition, is D of finite prime character? Question 5. If D has finite prime character and S is a multiplicatively closed subsetofd,doesd S havefiniteprimecharacter? R [1] D.D. Anderson, Globalization of some local properties in Krull domains, Proc. Amer. Math. Soc. 85 (1982), [2] D.D. Anderson, Star operations induced by overrings, Comm. Algebra 16(1988), [3] D.D. Anderson, D.J. Kwak and M. Zafrullah, Agreeable domains, Comm. Algebra 23(1995), [4] D.D. Anderson and L.A. Mahaney, Commutative rings in which every ideal is a product of primary ideals, J. Algebra 106(1987), [5] D.D. Anderson and L.A. Mahaney, On primary factorizations, J. Pure Appl. Algebra 54 (1988), [6] D.D. Anderson, J. Mott and M. Zafrullah, Finite character representations for integral domains, Bull. Math. Ital.(7) 6-B (1992), [7] D.D. Anderson and M. Roitman, A characterization of cancellation ideals, Proc. Amer. Math. Soc. 125 (1997), [8] D.D. Anderson and M. Zafrullah, Splitting sets and weakly Matlis domains, Commutative Algebra and Applications-Proceedings of the 2008 Fez Conference (to appear).

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