On finite molecularization domains
aa r X i v : . [ m a t h . A C ] J a n ON FINITE MOLECULARIZATION DOMAINS
ANDREW J. HETZEL, ANNA L. LAWSON, AND ANDREAS REINHART
Abstract.
In this paper, we advance an ideal-theoretic analogue of a “finitefactorization domain” (FFD), giving such a domain the moniker “finite molec-ularization domain” (FMD). We characterize FMD’s as those factorable do-mains (termed “molecular domains” in the paper) for which every nonzero idealis divisible by only finitely many nonfactorable ideals (termed “molecules” inthe paper) and the monoid of nonzero ideals of the domain is unit-cancellative,in the language of Fan, Geroldinger, Kainrath, and Tringali. We develop anumber of connections, particularly at the local level, amongst the conceptsof “FMD”, “FFD”, and the “finite superideal domains” (FSD’s) of Hetzel andLawson. Characterizations of when k [ X , X ], where k is a field, and the clas-sical D + M construction are FMD’s are provided. We also demonstrate that if R is a Dedekind domain with the finite norm property, then R [ X ] is an FMD. INTRODUCTION
Throughout this paper, all rings are commutative with 1 = 0. For over a century,the study of various types of decompositions of ideals has occupied an importantplace in commutative ring theory. While the famed Noether-Lasker theorem is of-ten considered the archetype for such investigations, invaluable research has beenconducted related to decomposing ideals as a product of a certain type of ideal; see,amongst many others, [24],[25], [29],[6],[3] (it should be noted that, in general, theterm “factoring” is used with regard to products of ideals while the term “decom-posing” is used with regard to intersections of ideals). Moreover, a modern focus onproducts of ideals has tremendous worth even in the classical context of algebraicgeometry that motivated the Noether-Lasker theorem. To wit, in light of the factthat the algebraic variety of a product of ideals is the same as the variety of thecorresponding intersection, computing a basis for a product of ideals is far morestraightforward than computing a basis for an intersection of ideals.In addition, while the idea of an “irreducible ideal” is quite natural for consid-erations of decomposing ideals, it was not until 1964 that a truly parallel notion,embodied by the concept of a “nonfactorable ideal”, for factoring ideals was ad-vanced by H.S. Butts [10]. A nonfactorable ideal I of a commutative ring R is a Mathematics Subject Classification.
Primary: 13A05; Secondary: 13A15, 13E05, 13F15,13F20.
Key words and phrases. atomic domain, factorable domain, finite factorization domain, fi-nite superideal domain, molecular domain, molecularization, molecule, nonfactorable ideal, unit-cancellative.This work is based in part on the second-named author’s master’s research at Tennessee TechUniversity.The third-named author of this work was supported by the Austrian Science Fund FWF,Project Number J4023-N35.The authors wish to express their thanks to the referee of this article for his thorough andthoughtful reading of the manuscript. nonzero proper ideal of R such that whenever I = JK for some ideals J and K of R , it must be the case that either J = R or K = R (see also [11]). In [10],Butts demonstrated that if R is an integral domain, then every nonzero, properideal of R can be factored uniquely (up to the order of the factors) as a productof nonfactorable ideals of R if and only if R is a Dedekind domain. As such, theconcept of “Dedekind domain” can be viewed as the proper ideal-theoretic analogueof “unique factorization domain”.Arguably inspired by the tremendous fruitfulness in studying certain general-izations of a “unique factorization domain” in [1], D.F. Anderson, H. Kim, and J.Park [7] introduced and explored factorable domains –domains R with the propertythat every nonzero proper ideal of R is a product of nonfactorable ideals of R –anideal-theoretic analogue to atomic domains. Consistent with this perspective, inthis paper, we wish to advance an ideal-theoretic analogue of the notion of a “finitefactorization domain” (or FFD), one of the generalizations of a “unique factoriza-tion domain” introduced in [1]. On our way to discovering definitive informationfor such an analogue in the contexts of certain polynomial rings (see Theorem 4.1,Corollary 4.9) and the classical D + M construction (see Theorem 5.2), we pick upsome novel information about factorable domains (notably Theorem 5.1) and evennonfactorable ideals themselves (notably Proposition 2.1, Proposition 4.2, Propo-sition 4.4, Theorem 4.5).At this point, a major caveat concerning terminology in this paper is warranted.In spite of the use in [10], [11], [7], [20], and [23] of the terms “nonfactorable ideal”,“factorable domain”, and “factorable ring”, we do not wish to continue this prac-tice in the present paper. The main reasons for this are that (1) the latter twoterms create an ambiguity if taken outside of context, as there are a fair numberof different types of factorizations of ideals (as mentioned above) that a term like“factorable” may reference, and (2) such terms do not seem to connect with theelement-level inspiration for these notions, where expressions such as “atom” and“atomic domain” are standard. To begin remedying these issues, we have cho-sen to adopt the terms “molecule” for “nonfactorable ideal”, “molecular domain”for “factorable domain”, and “molecularization” to mean “product of molecules”.Moreover, the term “finite ideal factorization domain” in [23] will henceforth bereplaced with “finite molecularization domain”, the focal concept of this paper. Inaddition, a nonzero proper ideal that is not a molecule will be called “compound”.Overall, such a change in vocabulary has the virtue of creating a certain idiomaticaesthetic, particularly in view of results such as Proposition 2.1, Corollary 2.2, andProposition 2.13 in this paper.As usual, the set of whole numbers (that is, the set of nonnegative integers)will be represented by W . Let R be a domain. The group of units of R will bedesignated by U ( R ). Throughout this paper, the dimension of R , denoted dim( R ),always refers to the Krull dimension of R , that is, the supremum of the lengths ofall chains of prime ideals of R . In particular, if P is a prime ideal of R , then the height of P is the dimension of the localization R P . If I is an ideal of R , then thenilradical of I is given by √ I = ∩{ P | P is a prime ideal of R containing I } . If I and J are ideals of R , we say that J divides I if there exists an ideal K of R forwhich I = JK .Furthermore, we distinguish between calling the domain R quasilocal if it hasa unique maximal ideal and local if additionally R is Noetherian. Similarly, we N FINITE MOLECULARIZATION DOMAINS 3 distinguish between calling R quasisemilocal if it has only finitely many maximalideals and semilocal if additionally R is Noetherian. The integral closure of R in some (given) field extension of the quotient field of R will be denoted by R ;where specification is not provided, the field extension may be assumed to be thequotient field itself. An overring of R is a ring containing R and contained withinthe quotient field of R . As in [19], R is called a finite superideal domain (or FSD)if every nonzero proper ideal of R has only finitely many ideals of R containing it.Any unexplained terminology is standard, as in [17], [21], [8].2. PROPERTIES OF MOLECULES
We begin this paper by presenting several novel properties of molecules. Ourinaugural results, Proposition 2.1 below and its associated Corollary 2.2, drawvaluable connections between the type of ideal-level factorizations being exploredin this paper and the corresponding type of element-level factorizations as a productof irreducibles. In particular, the relationship between molecules and atoms is mostintimate for those domains with trivial Picard groups (which include the respectiveclasses of quasisemilocal domains, B´ezout domains, UFD’s, and, thanks to [26,Theorem 6.1], one-dimensional domains with nonzero Jacobson radical).
Proposition 2.1.
Let R be a domain and I = ( a ) a principal ideal of R . If I is amolecule, then a is an atom. If R is further assumed to have trivial Picard group,then the converse is true, as well. Moreover, in this context, if J is an ideal thatdivides the principal ideal I , then J is also principal.Proof. Let R be a domain and let I = ( a ) be a principal ideal of R . If a is notan atom, then a = bc for some nonunits b and c , which implies I = ( b )( c ), where( b ) and ( c ) are proper ideals of R . Hence, I is not a molecule.Now, further suppose that R has trivial Picard group and that a is an atom.Assume to the contrary that I = JK for proper ideals J and K . Since J and K are necessarily invertible, it must be the case that J and K are principal. Thus, I = ( b )( c ), and so a = ubc for some unit u . However, the assumption that J and K are proper guarantees that b and c are nonunits, contradicting the irreducibilityof a . Therefore, it must be the case that I is a molecule.Finally, observe that even without the assumption that a is an atom, the abovework shows that both J and K are principal. Thus, any ideal that divides a prin-cipal ideal of a domain R with trivial Picard group is itself principal. ✷ Corollary 2.2.
A molecular domain with trivial Picard group is atomic.Proof.
Let R be a molecular domain with trivial Picard group and let a bea nonzero nonunit of R . Since R is molecular, there exists a molecularization( a ) = I I · · · I n . By Proposition 2.1 above, each I j is principal and, moreover, isgenerated by an atom a j of R . Hence, a = ua a · · · a n for some unit u , and so a can be written as a product of atoms, as desired. ✷ Recall that a multiplication ideal of a ring R is an ideal I of R satisfying theproperty that for any ideal J ⊆ I of R , there exists an ideal K of R for which J = IK . Note that an invertible ideal is a multiplication ideal. Proposition 2.3below, while elementary, highlights the intuitive idea that “multiplication ideal” and HETZEL, LAWSON, AND REINHART “molecule” are dynamically dissimilar in terms of the degree of ideal factorizationinvolved.
Proposition 2.3.
Let R be a domain and I a molecule of R . If J is a proper idealof R for which I ( J , then J cannot be a multiplication ideal of R .Proof. Deny. Then there exists a necessarily proper ideal K of R for which I = JK , contradicting the fact that I is a molecule of R . ✷ Certainly, the class of Dedekind domains reveals that sometimes the only ideals ofa domain that are molecules are the maximal ideals themselves. However, maximalideals need not be molecules, for instance, as in a valuation domain with a non-principal maximal ideal (in fact, such a domain has no molecules at all).It is natural then to seek out domains where a cancellation-type property forideals holds, as such contexts can give rise to the existence of molecules from whichan exploration of associated factorizations can begin. We find an abundance offruitfulness in considering domains that have what we deem “unit-cancellation forideals”. Motivated by the “pr´esimplifiable condition” for commutative rings withidentity ( xy = x ⇒ x = 0 or y is a unit), we say that an ideal I of the domain R is unit-cancellative if for each ideal J of R with I = IJ , it must be the case that J = R . By extension, we say the domain R has unit-cancellation for ideals if everynonzero ideal of R is unit-cancellative. This notion is equivalent to the monoid ofnonzero ideals of R being “unit-cancellative” in the terminology of [13].Propositions 2.4 and 2.6 below provide for a wealth of domains that have unit-cancellation for ideals. Recall that if S is a ring, R a subring of S , and I an idealof R , then we say that I survives in S if IS is a proper ideal of S and, moreover,we say that R ⊆ S is survival extension if every proper ideal of R survives in S .Clearly, every integral extension is a survival extension, owing to the lying-overtheorem. Also, if S is quasisemilocal and U ( S ) ∩ R = U ( R ), then an application ofprime avoidance reveals that R ⊆ S is a survival extension. Proposition 2.4.
Let Ω be a nonempty set of domains that have unit-cancellationfor ideals and let R be a domain that is a subring of each S ∈ Ω . If for eachproper ideal I of R there is some S ∈ Ω such that I survives in S , then R hasunit-cancellation for ideals.Proof. Let I be a nonzero ideal of R and J an ideal of R for which I = IJ .Assume that J is proper. Then JS = S for some S ∈ Ω. On the other hand, IS = ISJS , and hence JS = S , a contradiction. ✷ Corollary 2.5. If R ⊆ S is a survival extension of domains and S has unit-cancellation for ideals, then R has unit-cancellation for ideals. Proposition 2.6.
Let R be a domain. If R satisfies the conclusion of the Krullintersection theorem (that is, ∩ ∞ n =0 M n = 0 for all maximal ideals M of R ), then R has unit-cancellation for ideals. In particular, if R is Noetherian or the integralclosure R in some field extension of the quotient field of R is Noetherian, then R has unit-cancellation for ideals.Proof. Let R be a domain that satisfies the conclusion of the Krull intersectiontheorem. Let I be a nonzero ideal of R and J an ideal of R for which I = IJ . N FINITE MOLECULARIZATION DOMAINS 5
Observe that I = IJ n for every whole number n , and so I ⊆ ∩ ∞ n =0 J n . We concludethat J = R . Therefore, R has unit-cancellation for ideals.Since it is well-known that every Noetherian domain satisfies the conclusion ofthe Krull intersection theorem, it now follows that if R is a Noetherian domain,then R has unit-cancellation for ideals (an alternative justification of the fact thatNoetherian domains have unit-cancellation for ideals is found by observing that anynonzero finitely generated ideal of a domain is unit-cancellative, an application ofNakayama’s lemma being all that is necessary to show this). Suppose then that theintegral closure R in some field extension of the quotient field of R is Noetherian andlet M be a maximal ideal of R . By the lying-over theorem, there is some maximalideal M of R such that M ∩ R = M . Since R satisfies the conclusion of the Krullintersection theorem, we have that ∩ ∞ n =0 M n ⊆ ∩ ∞ n =0 M n = 0, and so ∩ ∞ n =0 M n = 0.Therefore, R itself must satisfy the conclusion of the Krull intersection theoremfrom which we deduce that R itself has unit-cancellation for ideals. The proof isthus complete. ✷ It should be noted that a polynomial ring in infinitely many indeterminates overa field has unit-cancellation for ideals, but the ring is not Noetherian nor is anyintegral extension of the ring Noetherian.We now give a proposition (Proposition 2.7) that characterizes molecules interms of the unit-cancellation property. As a consequence, domains with unit-cancellation for ideals adequately address an issue mentioned in the discussion justprior to Proposition 2.6.
Proposition 2.7.
Let R be a domain, I a proper ideal of R , P a prime ideal of R , and M a maximal ideal of R .(a) I is a molecule of R if and only if (1) I is unit-cancellative and (2) for allideals J and K of R for which I = JK , it follows that either J = I or K = I .(b) P is a molecule if and only if P is unit-cancellative.(c) (cf. [20, Corollary 2.3] ) M is a molecule if and only if M is not idempotent.Proof. (a) Trivially, if I is a molecule of R , then I is unit-cancellative and forall ideals J and K of R such that I = JK , it must be the case that either J = I or K = I . Conversely, let J and K be ideals of R such that I = JK . Then either J = I or K = I by hypothesis. Without loss of generality, assume that J = I .Then I = IK , and so K = R since I is unit-cancellative. Therefore, I is a moleculeof R .(b) Clearly, if J and K are ideals of R such that P = JK , then either J = P or K = P . Therefore, the statement is an immediate consequence of (a).(c) If M is unit-cancellative, then M is clearly not idempotent. If M is not idem-potent and J is an ideal of R such that M = M J , then M is properly contained in J , whence J = R , and so M is unit-cancellative. The statement now follows from(b). ✷ Proposition 2.8 below reveals that molecules share a property of strongly ir-reducible ideals in regards to products of pairwise comaximal ideals. It is thisconsideration of products of a certain type of ideal that also gives rise to a largeclass of domains where every molecule is a primary ideal (cf. [20, Example 2.14]),formalized in Proposition 2.9.
HETZEL, LAWSON, AND REINHART
Proposition 2.8.
Let R be a domain and I a molecule of R . Let J , J , . . . , J n bepairwise comaximal ideals of R such that J J · · · J n ⊆ I . Then J i ⊆ I for some i = 1 , , . . . , n .Proof. Note that if J , J , . . . , J n are pairwise comaximal ideals of R , then P \ J J · · · J n = R . As such I = ( I + J )( I + J ) · · · ( I + J n ). However, since I is amolecule, I + J i = I for some i = 1 , , . . . , n . This means that J i ⊆ I , as desired. ✷ Proposition 2.9.
Let R be a Laskerian domain such that every non-maximal primeideal of R is a multiplication ideal of R . Then every molecule of R is a primaryideal of R .Proof. By [2, Theorem 10], every proper ideal of R is a finite product of primaryideals of R . It is obvious then that every molecule of R is a primary ideal of R . ✷ We can also specialize to sufficient conditions on a molecule itself that guaranteethat the molecule is primary, as given in Proposition 2.10 and Corollary 2.11 below.This has the upshot of providing an alternative means of obtaining a result onmolecules of a one-dimensional Noetherian domain (Corollary 2.12) that is also aconsequence of Proposition 2.9.
Proposition 2.10.
Let R be a domain and I a molecule of R such that R/I isa quasisemilocal zero-dimensional ring whose nilradical is nilpotent. Then I is aprimary ideal of R .Proof. Since
R/I is quasisemilocal and zero-dimensional, we have that the setof all prime ideals of R that contain I is finite and consists only of maximal idealsof R . Let { M , M , . . . , M n } be the set of all prime ideals of R that contain I .Since the nilradical of R/I , which is √ I/I , is nilpotent, it follows that ( √ I ) m ⊆ I for some m ∈ N . However, this means that Q ni =1 M mi ⊆ I , and so M mj ⊆ I forsome j = 1 , , . . . , n by Proposition 2.8. We conclude then that I is M j -primary. ✷ Corollary 2.11.
Let R be a domain and I a molecule of R such that R/I isArtinian. Then I is a primary ideal of R . Corollary 2.12.
Let R be a one-dimensional Noetherian domain. Then everymolecule of R is a primary ideal of R . Proposition 2.13 below provides one of the key reasons for opting for the “mol-ecule” terminology over the previous “factorable” language. In particular, notingButts’ characterization of a domain with unique factorization of ideals into non-factorable ideals as a Dedekind domain [10, Theorem], one should recognize theparallel of the statement of Proposition 2.13 with the well-known characterizationof UFD’s as atomic domains where every atom is prime.
Proposition 2.13.
Let R be a domain. Then R is a molecular domain for whichevery molecule is prime if and only if R is a Dedekind domain.Proof. The fact that if R is a Dedekind domain, then R is a molecular domainfor which every molecule is prime was established by Butts [10]. Conversely, if R is a molecular domain for which every molecule is prime, then every ideal of R isa product of prime ideals of R , and it is well-known that such a domain must be N FINITE MOLECULARIZATION DOMAINS 7
Dedekind. ✷ FINITE MOLECULARIZATION DOMAINS
We now come to the main concept of this paper, the notion of a “finite molecu-larization domain”, an idea designed to be the ideal-theoretic analogue of a “finitefactorization domain”.
Definition 3.1. A finite molecularization domain , or FMD, is a molecular domain R with the property that every nonzero proper ideal of R has only a finite number ofmolecularizations–that is, every nonzero proper ideal of R has only a finite numberof factorizations as a product of molecules. Moreover, if I is an ideal of R suchthat I = J J · · · J m and I = K K · · · K n , where each J i and K j is a moleculeof R , then these factorizations will be regarded as the same if m = n and there isa permutation σ ∈ S n for which J i = K σ ( i ) for i = 1 , , . . . , n .In light of the obvious connection that the notion of “finite molecularizationdomain” has with “finite factorization domain”, it is natural to ask if the property ofbeing an FMD can be characterized, at least in part, by the finiteness of the numberof (non-associated) molecular divisors (see [1, p. 2]). Corollary 3.3 below reveals anaffirmative answer to this question. In fact, the characterization presented throughCorollary 3.3 is powerful enough to spawn several corollaries of its own, two of which(Corollaries 3.6 and 3.4) establish that every FSD is an FMD and every FMD is anFFD.We first lay the groundwork for Corollary 3.3 through Theorem 3.2, which ad-dresses the situation at the level of the ideals themselves. Theorem 3.2.
Let R be a molecular domain and I a unit-cancellative ideal of R that is divisible by only finitely many molecules of R . Then I has only a finitenumber of molecularizations (that is, I has only a finite number of factorizationsas a product of molecules of R ) and I is divisible by only finitely many ideals of R .Proof. Let R be a molecular domain and I a unit-cancellative ideal of R that isdivisible by only finitely many molecules of R . Without loss of generality, we mayassume that I is a nonzero proper ideal of R . Let { J , J , . . . , J n } be the set of allmolecules that divide I . Put E = { ( α , α , . . . , α n ) ∈ W n | I = Q ni =1 J α i i } . ThenMin( E ) is finite by Dickson’s lemma.Next, we show that Min( E ) = E . Let α = ( α , α , . . . , α n ) ∈ E . By Dickson’slemma, there is some β = ( β , β , . . . , β n ) ∈ Min( E ) such that β ≤ α (that is, β i ≤ α i for each i = 1 , , . . . , n ). Thus, I = Q ni =1 J α i i = Q ni =1 J β i i Q ni =1 J α i − β i i = I Q ni =1 J α i − β i i and hence Q ni =1 J α i − β i i = R . As such, α = β , and so Min( E ) = E .It follows that E is finite, and thus I has only a finite number of factorizations asa product of molecules of R . Furthermore, observe that the set of ideals of R thatdivide I is given by { Q ni =1 J γ i i | γ = ( γ , γ , . . . , γ n ) ∈ W n and γ ≤ α for some α ∈ E } , which is clearly finite. The proof is thus complete. ✷ Corollary 3.3.
Let R be a domain. The following are equivalent:(1) R is an FMD;(2) R is a molecular domain, R has unit-cancellation for ideals, and everynonzero ideal of R is divisible by only finitely many molecules of R ; HETZEL, LAWSON, AND REINHART (3) R has unit-cancellation for ideals and every nonzero ideal of R is divisibleby only finitely many ideals of R .Proof. (1) ⇒ (2): Obviously, R is molecular. Let I be a nonzero proper ideal of R .Clearly, I is divisible by only finitely many molecules of R . Let { J , J , . . . , J n } bethe set of all molecules that divide I . Since R is an FMD, E = { ( γ , γ , . . . , γ n ) ∈ W n | I = Q ni =1 J γ i i } is finite. In particular, there is some α = ( α , α , . . . , α n ) ∈ W n such that γ ≤ α for all γ ∈ E . Let J be an ideal of R such that I = IJ . Thenthere are β = ( β , β , . . . , β n ) ∈ W n and γ = ( γ , γ , . . . , γ n ) ∈ W n for which I = Q ni =1 J β i i and J = Q ni =1 J γ i i . However, if m ∈ W , then I = IJ m , whence I = Q ni =1 J β i + mγ i i . But this means that β + mγ ≤ α for every m ∈ W , and so γ i = 0 for each i = 1 , , . . . , n . Therefore, J = R , and we conclude that I isunit-cancellative.(2) ⇒ (1) and (3): This follows immediately from Theorem 3.2 above.(3) ⇒ (2): It is sufficient to show that R is a molecular domain. Let I be anonzero proper ideal of R . Assume to the contrary that I is not a finite product ofmolecules of R . Clearly, there is a proper ideal J of R that divides I such that J is maximal amongst the proper ideals of R that divide I and that are not a finiteproduct of molecules of R . Since J itself cannot be a molecule of R , it follows byProposition 2.7 that there are proper ideals A and B of R such that J ( A, B and J = AB . But then A and B are each finite products of molecules of R , whence J must be too, a contradiction. This completes the proof. ✷ Corollary 3.4.
Every FMD is an FFD.Proof.
From Corollary 3.3, it follows that every nonzero principal ideal of theFMD R is contained in only finitely many principal ideals of R , and thus R is anFFD by [4, Theorem 1]. ✷ Corollary 3.5.
Let R be a Noetherian domain. Then R is an FMD if and only ifevery nonzero ideal of R is divisible by only finitely many molecules of R . Corollary 3.6.
Every FSD is an FMD.Proof.
Let R be an FSD. Then R is Noetherian [19, Proposition 2.1] and ev-ery nonzero ideal of R is contained in (and thus divisible by) only finitely manymolecules of R . Therefore, R is an FMD by Corollary 3.5. ✷ In light of Corollaries 3.5 and 3.6 above, it should be noted that while everyNoetherian domain is molecular (see [20, Theorem 2.13]), not every Noetheriandomain is an FMD, as revealed in Theorems 4.1 and 5.2. In fact, Theorem 5.2provides for the existence of one-dimensional local domains that are not FMD’s.Nonetheless, since it is well-known that FFD’s satisfy ACCP, Corollary 3.4 givesthat FMD’s also satisfy ACCP.It also should be pointed out that the converse of the implication in Corollary3.4 is false, in general. Since [7, Theorem 5] implies that a Pr¨ufer FMD is aDedekind domain, any one-dimensional Pr¨ufer FFD that is not Dedekind (see,amongst others, [18, Example 2]) cannot be an FMD.Now, given that the property of unit-cancellation for ideals and the property thatevery nonzero ideal is divisible by only finitely many ideals were used to characterize
N FINITE MOLECULARIZATION DOMAINS 9
FMD’s in Corollary 3.3, we offer the following result in the spirit of the domainextension considerations of Proposition 2.4 and Corollary 2.5.
Proposition 3.7.
Let S be a domain and R a subring of S such that the conductorideal ( R : S ) = 0 . If every nonzero ideal of R is divisible by only finitely manyideals of R , then every nonzero ideal of S is divisible by only finitely many idealsof S .Proof. Let I be a nonzero ideal of S and let x ∈ ( R : S ) be nonzero. Observethat if J and L are ideals of S such that I = JL , then x I = xJxL and x I , xJ ,and xL are ideals of R . Let I be the set of ideals of S that divide I and let J bethe set of ideals of R that divide x I . Let f : I → J be defined by f ( J ) = xJ .Since f is a well-defined injective map, we must have that I is finite, as desired. ✷ We now transition to local-global considerations with regards to the propertyof being an FMD. Our main result along these lines, Theorem 3.9, shows that theproperty of being an FMD is stable at localizations of height-one maximal ideals (ascontrasted with the property of being an FFD which is not, in general, stable underthe formation of localizations at height-one maximal ideals; see [18, Example 2] and[1, Example 5.4]). We first provide a lemma regarding some special properties ofideals of localizations at height-one maximal ideals.
Lemma 3.8.
Let R be a domain, M a height-one maximal ideal of R , and I anideal of R M .(a) If C and D are ideals of R M such that I = CD , then I ∩ R = ( C ∩ R )( D ∩ R ) .(b) If I is principal, then I ∩ R is locally principal.(c) I is the only ideal of R M whose contraction to R is I ∩ R .Proof. (a) Let C and D be ideals of R M such that I = CD . Without loss ofgenerality, we may assume that I is a nonzero proper ideal of R M . Put A = C ∩ R and B = D ∩ R . Since I is a primary ideal of R M , we have that I ∩ R is an M -primaryideal of R . Moreover, I ∩ R ⊆ A ∩ B , and thus M = √ I ∩ R ⊆ √ A ∩ √ B = √ AB .If √ AB = R , then A = B = R , whence I = R M , a contradiction. Therefore, √ AB = M , and so AB is M -primary. Since C = A M and D = B M , we concludethat I ∩ R = ( AB ) M ∩ R = AB = ( C ∩ R )( D ∩ R ).(b) Let I be principal and nonzero, and let N be a maximal ideal of R . Notethat M ⊆ √ I ∩ R . If N = M , then I ∩ R is not contained in N , and hence( I ∩ R ) N = R N is principal. On the other hand, if N = M , then ( I ∩ R ) N = I isprincipal.(c) Let J be an ideal of R M such that J ∩ R = I ∩ R . Then J = ( J ∩ R ) M =( I ∩ R ) M = I . ✷ Theorem 3.9.
Let R be a domain and M a height-one maximal ideal of R .(a) If R has unit-cancellation for ideals, then R M has unit-cancellation for ideals.(b) If every nonzero ideal of R is divisible by only finitely many ideals of R , thenevery nonzero ideal of R M is divisible by only finitely many ideals of R M .(c) If R is an FMD, then R M is an FMD.Proof. (a) Let I be a nonzero ideal of R M and J an ideal of R M such that I = IJ . By Lemma 3.8(a), we have that I ∩ R = ( I ∩ R )( J ∩ R ). Clearly, I ∩ R is nonzero, and thus I ∩ R is a unit-cancellative ideal of R . It follows that J ∩ R = R ,and so J = R M .(b) Let I be a nonzero ideal of R M . Let I be the set of ideals of R M thatdivide I and J the set of ideals of R that divide I ∩ R . Let f : I → J be given by f ( L ) = L ∩ R . Then f is a well-defined injective map by Lemma 3.8. Since J isfinite, we must have that I is finite.(c) This is an immediate consequence of (a) and (b) above and Corollary 3.3. ✷ Note that a certain converse of Theorem 3.9(a) is true, in the sense that if R M has unit-cancellation for ideals for every maximal ideal M of R , then R has unit-cancellation for ideals, thanks to Proposition 2.4.While we can certainly deduce from Theorem 3.9 that if R is an FMD and M is aheight-one maximal ideal of R , then R M is an FFD (see Corollary 3.4), Proposition3.10 below shows that this conclusion can actually be obtained under a slightlymore general hypothesis. Proposition 3.10.
Let R be a domain for which each nonzero locally principalideal is divisible by only finitely many locally principal ideals. If M is a height-onemaximal ideal of R , then R M is an FFD.Proof. Let I be a nonzero principal ideal of R M . Let I be the set of principalideals of R M that divide I and J the set of locally principal ideals of R that divide I ∩ R . Let f : I → J be given by f ( L ) = L ∩ R . Then f is a well-defined injectivemap by Lemma 3.8. Since J is finite, we must have that I is finite. Since principalideals are multiplication ideals, there are only finitely many principal ideals of R M that contain I . Therefore, R M is an FFD by [4, Theorem 1]. ✷ We conclude this section making contact again with the FSD property. In par-ticular, after presenting a novel characterization through Proposition 3.11 below ofwhen the property of being an FSD globalizes, we are able to present in Theorem3.13 a characterization of FSD’s in terms of the FMD property. As a consequenceof this theorem (Corollary 3.14), we are able to answer a conjecture presented in[19] regarding sufficient conditions guaranteeing the equivalence of the FSD andFFD properties.
Proposition 3.11.
Let R be a domain. Then R is an FSD if and only if R is offinite character and locally an FSD.Proof. Let R be an FSD. Clearly, R is of finite character. It also follows by [19,Theorem 3.5] that R M is an FSD for each maximal ideal M of R .Conversely, let R be of finite character and locally an FSD. Let I be a nonzeroproper ideal of R . Let M , M , . . . , M n be the maximal ideals of R that contain I .For each i = 1 , , . . . , n , let J i be the set of ideals of R M i that contain I M i and let J be the set of ideals of R that contain I . If J ∈ J and Max( R ) is the set of allmaximal ideals of R , then J = R ∩ (cid:16) ∩ M ∈ Max ( R ) J M (cid:17) = R ∩ ( ∩ ni =1 J M i ). Therefore, f : Q ni =1 J i → J defined by f (( J i ) ni =1 ) = R ∩ ( ∩ ni =1 J i ) is a well-defined surjectivemap. However, since J i is finite for each i = 1 , , . . . , n , it must be the case that J is finite, as desired. ✷ N FINITE MOLECULARIZATION DOMAINS 11
We pause briefly to expand upon the value of the “finite character” hypothesisutilized in Proposition 3.11 above by recognizing its worth for FMD considerationsthrough Proposition 3.12 below.
Proposition 3.12.
Let R be a domain of finite character.(a) If for each maximal ideal M of R we have that every nonzero ideal of R M is divisible by only finitely many ideals of R M , then every nonzero ideal of R isdivisible by only finitely many ideals of R .(b) If R is locally an FMD, then R is an FMD.Proof. (a) Let I be a nonzero ideal of R . Let I be the set of all ideals of R that divide I , and for each maximal ideal M of R , let I M be the set of all idealsof R M that divide I M . Let Max( R ) be the set of all maximal ideals of R , and let f : I → Q M ∈ Max ( R ) , I ⊆ M I M be given by f ( L ) = ( L M ) M ∈ Max ( R ) , I ⊆ M . Clearly, f is a well-defined map. Let J, L ∈ I be such that f ( J ) = f ( L ) and let M ∈ Max( R ).If I ⊆ M , then we clearly have that J M = L M . If I * M , then J * M and L * M ,and so J M = R M = L M . We deduce that J = L , and thus f is injective. It followsthat I is finite, as desired.(b) This follows from (a) above, Proposition 2.4, and Corollary 3.3. ✷ We note that the “finite character” hypothesis cannot be dropped from Proposi-tion 3.12 above by considering an almost Dedekind domain R that is not a Dedekinddomain. For such a domain R is locally an FSD, hence locally an FMD, but, since R is a Pr¨ufer domain that is not a Dedekind domain, it cannot be an FMD (see [7,Theorem 5]).We now give the promised characterization of FSD’s in terms of the FMD prop-erty. Theorem 3.13.
Let R be a domain. The following are equivalent:(1) R is an FSD;(2) R is a Noetherian FMD and dim ( R ) ≤ ;(3) R is Noetherian, dim ( R ) ≤ , and every invertible ideal of R is contained inonly finitely many invertible ideals of R .Proof. (1) ⇒ (2): Let R be an FSD. By [19, Proposition 2.1], we have that R isNoetherian and dim( R ) ≤
1. It follows from Corollary 3.6 that R is an FMD, aswell.(2) ⇒ (3): Let I be an invertible ideal of R . By Corollary 3.3, I is divisible byonly finitely many (invertible) ideals of R . The assertion now follows since invertibleideals are multiplication ideals.(3) ⇒ (1): Without loss of generality, we may assume that R is not a field.Since R is Noetherian, we have that every nonzero locally principal ideal of R isinvertible. If M is a maximal ideal of R , then R M is Noetherian, and hence R M is an FSD by combining Proposition 3.10 with [19, Theorem 2.5]. Clearly, R is offinite character, and so R must be an FSD by Proposition 3.11. ✷ Corollary 3.14.
Let R be a Noetherian one-dimensional domain whose Picardgroup is trivial. Then R is an FSD if and only if R is an FFD. As a result of Corollary 3.14 above, the first conjecture expressed in the remarkimmediately following [19, Theorem 2.5] is proved. In particular, if R is a one-dimensional (Noetherian) semilocal domain, then R is an FSD if and only if R isan FFD. Moreover, this equivalence breaks down if the “semilocal” hypothesis isremoved, as evidenced by the fact that k [ X , X ], where k is an infinite field, isa one-dimensional Noetherian FFD that is not an FSD (see Theorem 4.1 and [4,Theorem 3]).Let R be a domain. Recall that R is called seminormal if whenever x is anelement of the quotient field of R such that x , x ∈ R , it must be the case that x ∈ R , and the seminormalization of R is the intersection of all seminormal overringsof R . We will denote the complete integral closure of R by b R . As in [17], we saythat R is a G -domain if the intersection of all nonzero prime ideals of R is nonzero.Our final result of this section, Theorem 3.15, improves upon the characterizationof FSD’s offered in Theorem 3.13 under the additional hypothesis that ( R : S ) = 0,where S is the seminormalization of R . Theorem 3.15.
Let R be a domain and S the seminormalization of R . If theconductor ideal ( R : S ) = 0 , then the following are equivalent:(1) R is an FSD;(2) R is an FMD and R is a Dedekind domain;(3) R is a Mori domain, dim ( R ) ≤ , and every invertible ideal of R is containedin only finitely many invertible ideals of R ;(4) dim ( R ) ≤ , b R is a Krull domain, and every nonzero locally principal idealof R is divisible by only finitely many locally principal ideals of R .Proof. We begin by claiming that if ( R : S ) = 0, where S is the seminormaliza-tion of R , then ( R P : c R P ) = 0 for each height-one prime ideal P of the domain R .For let P be a height-one prime ideal of the domain R , let S be the seminormal-ization of R , and suppose that ( R : S ) = 0. Note that ( R P : S P ) ⊇ ( R : S ) and S P ⊆ R P = R P ⊆ c R P ⊆ c S P . Since R P is quasilocal and one-dimensional, it mustbe a G -domain. We have that S P is an overring of R P , and thus S P is a G -domain,as well. Moreover, S P is seminormal, and so it follows by [15, Proposition 4.8.2]that ( S P : c R P ) ⊇ ( S P : c S P ) = 0. We conclude that ( R P : c R P ) ⊇ ( R P : S P )( S P : c R P ) = 0, and the claim is thus proved.(1) ⇒ (2) and (3): This follows immediately from Theorem 3.13 and the Krull-Akizuki theorem.(2) ⇒ (4): It follows from Corollary 3.3 that every nonzero ideal of R is divisibleby only finitely many ideals of R , and so the corresponding statement about locallyprincipal ideals of R is clearly true. Observe that dim( R ) = dim( R ) ≤
1. Since R is Noetherian, we have that R ⊆ b R ⊆ b R = R . Consequently, b R is a Dedekinddomain, and hence a Krull domain.(3) ⇒ (1): Without loss of generality, we may assume that R is not a field. Since R is a one-dimensional Mori domain, it is clear that R is of finite character.Next, we show that every nonzero locally principal ideal of R is invertible. Let I be a nonzero locally principal of R and N a maximal ideal of R . Since R is a Moridomain, there is some nonzero finitely generated ideal J of R such that J ⊆ I and I − = J − . Therefore, ( I − ) N = ( J − ) N = J − N ⊇ I − N ⊇ ( I − ) N . We deduce that( II − ) N = I N I − N = R N , and hence II − = R . N FINITE MOLECULARIZATION DOMAINS 13
Now, let M be a maximal ideal of R . By Proposition 3.11, it suffices to showthat R M is an FSD. Observe that R M is a Mori domain. The claim proved at thebeginning of this proof implies that ( R M : d R M ) = 0. By Proposition 3.10, R M isan FFD, and thus U ( d R M ) /U ( R M ) is finite by [4, Theorem 4]. It follows by [28,Theorem 4.2] that R M is Noetherian, and thus R M is an FSD by [19, Proposition2.16].(4) ⇒ (1): Without loss of generality, we may assume that R is not a field. Let M be a maximal ideal of R . By Proposition 3.10, R M is an FFD and, by the claimproved at the beginning of this proof, we have that ( R M : d R M ) = 0. Observe that b R M is a Krull domain and b R M ⊆ d R M . Since R M ⊆ b R M and b R M is completelyintegrally closed, we have that d R M ⊆ b R M . Therefore, d R M = b R M is a Krull domain.Since R M is a G -domain, d R M is a G -domain, and thus d R M is a semilocal PID (cf.[28, Lemma 3.1]). We conclude by [16, Lemma 5.2(1) and Proposition 5.6(2)] that R M is Noetherian, and thus R M is an FSD by [19, Proposition 2.16].Since ( R M : d R M ) = 0, we have that d R M is not a field. Let Q be a maximal idealof d R M . Then Q ∩ R M = M M . Since b R ⊆ d R M , it follows that Q ∩ b R is a nonzeroprime ideal of b R . Since b R is a Krull domain, there is some height-one prime ideal P of b R such that P ⊆ Q ∩ b R . We conclude that 0 = P ∩ R ⊆ Q ∩ R = Q ∩ R M ∩ R = M M ∩ R = M , and so P ∩ R = M . In particular, for each maximal ideal N of R ,there is a height-one prime ideal N ′ of b R such that N ′ ∩ R = N . Since each nonzeroelement of b R is contained in only finitely many height-one prime ideals of b R , wehave that R is of finite character. It follows by Proposition 3.11 that R is an FSD. ✷ RINGS OF POLYNOMIALS
We now turn our attention to investigating the FMD property as it pertains tocertain rings of polynomials. Motivated by the original algebraic geometric contextfor primary decompositions, we find complete information with respect to the ring k [ X , X ], that is, the ring of all polynomials over the field k that lack a linearterm, in Theorem 4.1. We then focus on standard polynomial rings R [ X ], where R is a domain and X an indeterminate over R , and discover in Corollary 4.9 thatsuch a ring is an FMD when R is a special type of Dedekind domain. Theorem 4.1.
Let k be a field. Then the following are equivalent:(1) k [ X , X ] is an FMD;(2) k [ X , X ] is an FSD;(3) k is finite.Proof. (1) ⇔ (2): Since every FSD in an FMD by Corollary 3.6, we immediatelyhave that (2) ⇒ (1). Conversely, suppose (1). Clearly, k [ X , X ] is a Noetherianone-dimensional domain. Consequently, k [ X , X ] is an FSD by Theorem 3.13.(2) ⇔ (3): Suppose (2). Then the ideal ( X ) is contained in only finitely manyideals of k [ X , X ], whence the set { ( X + bX , X ) | b ∈ k } is finite. It is sufficientthen to show that there is a bijection between k and { ( X + bX , X ) | b ∈ k } .To see this, let a, b ∈ k and suppose that ( X + bX , X ) = ( X + aX , X ).Then for some f, g ∈ k [ X , X ] it must be the case that X + bX = f X + gX + gaX ⇒ X (1 − g − f X ) = ( ga − b ) X ⇒ X | ( ga − b ) X . So, the X term of( ga − b ) X must be 0. Thus, g a = b , where g is the constant term of g . However, X | (1 − g ) X − f X and so it must be the case that the X term is 0. Thus, g = 1, and so a = b . The bijection is thus established.Conversely, suppose (3). Put T = k [ X ] and put R = k [ X , X ]. Then T isclearly integral over R and T is a finitely generated R -module. Since T has thefinite norm property, R is an FSD by [19, Theorem 3.6]. ✷ We next set out to investigate which ideals I of a general polynomial domain R [ X ] are molecules, not only towards a general understanding of such matters, butto discover sufficient conditions for a polynomial domain to be an FMD. Proposition4.2 and its associated Corollary 4.3 below reveal that the context of UFD’s allowsfor some definitive information along these lines.Recall that for an ideal I of the domain R , the t -closure of I is given by I t = ∪{ J v | J ⊆ I is a finitely generated ideal of R } , where J v = ( J − ) − . An ideal iscalled a t -ideal if it coincides with its t -closure. Note that every nonzero principalideal is a t -ideal and, moreover, the domain R is a UFD if and only if every t -idealof R is principal. Furthermore, if S ⊆ R \{ } is a multiplicatively closed subset of R and J is a t -ideal of the quotient overring S − R , then J ∩ R is a t -ideal of R . Proposition 4.2.
Let R be a UFD and I a nonzero ideal of R . If I ( I t ( R ,then I is a compound ideal of R .Proof. Let I ( I t ( R . Then I is properly contained in the proper multiplicationideal I t , and thus I is not a molecule of R by Proposition 2.3. ✷ Corollary 4.3.
Let R be a UFD and I a nonprincipal ideal of R [ X ] . If I ∩ R = 0 ,then I is a compound ideal.Proof. Let I ∩ R = 0 and let K be the quotient field of R . Observe that R [ X ]is a UFD and IK [ X ] ∩ R [ X ] ( R [ X ]. Since IK [ X ] is a principal ideal of K [ X ]and K [ X ] is a quotient overring of R [ X ], we have that IK [ X ] ∩ R [ X ] is a t -ideal of R [ X ]. This implies that I t ⊆ ( IK [ X ] ∩ R [ X ]) t = IK [ X ] ∩ R [ X ]. We conclude that I ( I t ( R [ X ], and hence I is not a molecule by Proposition 4.2. ✷ While Proposition 4.2 above established that certain ideals in a UFD are com-pound, Proposition 4.4 below provides for a wealth of ideals in an arbitrary domainthat are molecules.
Proposition 4.4.
Let R be a domain and let P and Q be nonzero locally principalideals of R . If P is a prime ideal of R and P + Q is a maximal ideal of R , then P + Q n is a molecule of R for each n ∈ N .Proof. Let P be a prime ideal of R , P + Q a maximal ideal of R , and n ∈ N .Put M = P + Q . If Q ⊆ P , then M = P + Q n = P , and thus P + Q n is a moleculeof R by Proposition 2.7(c). Therefore, we can assume that Q * P . Suppose to thecontrary that P + Q n is not a molecule of R . Then there are proper ideals J and K of R for which P + Q n = JK . It follows that M = √ P + Q n = √ J ∩ √ K , andhence √ J = √ K = M . As such, J and K are contained in M . We conclude that P ⊆ P + Q n = JK ⊆ M = P + P Q + Q . However, since P and Q are bothlocally principal, there exist p, h ∈ R M such that P M = pR M and Q M = hR M .Thus, we have that pR M = P M ⊆ P M + P M Q M + Q M = p R M + phR M + h R M ,hence there are g , g , g ∈ R M such that p = p g + phg + h g . Observe that N FINITE MOLECULARIZATION DOMAINS 15 h g ∈ P M . If h ∈ P M , then Q ⊆ Q M ∩ R ⊆ P M ∩ R = P , a contradiction. Wemay conclude that h P M , and so g = pg for some g ∈ R M . But this impliesthat 1 = pg + hg + h g ∈ P M + Q M = M M , a contradiction. Therefore, P + Q n is a molecule of R . ✷ Theorem 4.5 reveals that if R is further assumed to be a Dedekind domain, acharacterization along the lines of Proposition 4.4 is available. Theorem 4.5.
Let R be a Dedekind domain, P a nonzero prime ideal of R , and I a proper ideal of R [ X ] such that I ) P [ X ] . Then I is a molecule of R [ X ] if andonly if there are some n ∈ N and f ∈ R [ X ] such that P [ X ] + f R [ X ] is a maximalideal of R [ X ] and I = P [ X ] + f n R [ X ] .Proof. Observe that R [ X ] is a two-dimensional Noetherian domain and P [ X ] isan invertible height-one prime ideal of R [ X ]. Therefore, every prime ideal of R [ X ]that contains I is a maximal ideal of R [ X ].Suppose that I is a molecule of R [ X ]. It follows from Corollary 2.11 that I is pri-mary. Put M = √ I . Then M is a maximal ideal of R [ X ]. Note that R [ X ] /P [ X ] ∼ =( R/P )[ X ] is a PID and M/P [ X ] is a maximal ideal of R [ X ] /P [ X ]. Consequently, M = P [ X ] + f R [ X ] for some f ∈ M . Moreover, p I/P [ X ] = M/P [ X ], and so I/P [ X ] = ( M/P [ X ]) n for some n ∈ N . Therefore, I = P [ X ] + M n = P [ X ] + f n R [ X ], as desired.Since the converse is an immediate consequence of Proposition 4.4, the proof iscomplete. ✷ We pause briefly to note through Proposition 4.6 that, even if R is a PID, thereexist nonprincipal molecules of R [ X ] other than those of the form P [ X ] + f n [ X ],where P is a nonzero prime ideal of R and P [ X ] + f R [ X ] is a maximal ideal of R [ X ]. However, as the proof of Proposition 4.6 makes clear, demonstrating thatsuch ideals are molecules can prove to be relatively involved. Proposition 4.6.
Let R be a PID and p a prime element of R . The ideal ( X , p ) in R [ X ] is a molecule.Proof. Suppose to the contrary that P = ( X , p ) is compound, so that P = IJ for proper ideals I and J . Then P = ( P : J ) J , where P : J is also proper, owingto the fact that R [ X ] is a Noetherian domain. Let M be the maximal ideal ( X, p )in R [ X ] and observe that M ⊆ P , so that P : J and J are both contained in M .We will show that if P : J = M , then J = M .Suppose that P : J = M . Since M is maximal, it must be the case that M J * P .Since M J ⊆ M = ( X , pX, p ) and M J * P = ( X , p ), there exists n ∈ R suchthat gcd( n, p ) = 1 and npX ∈ M J − P . Let u, v ∈ R be such that 1 = un + vp .Then pX = u ( npX ) + vp X ∈ M J . Thus,
M J = M .Next, we show that J = M . Note that it suffices to show that p, X ∈ J . Since pX ∈ M J = XJ + pJ , there exist α, β ∈ J such that pX = Xα + pβ . Since p | α and X | β , it follows that α = hp and β = gX for some h, g ∈ R [ X ]. Thus, pX = hpX + gXp = pX ( h + g ), whence h + g = 1. However, since h = b + Xf for some b ∈ R and f ∈ R [ X ], it must be the case that bp + pXf = α ∈ J , and so bp ∈ J . Moreover, (1 − b ) X − X f = β ∈ J , and hence (1 − b ) X ∈ J . We thenhave the following two cases: Case I . Suppose that p | R b . Then bX ∈ J . Thus, X = bX + (1 − b ) X ∈ J .Since p ∈ XJ + pJ , there exist y, z ∈ J such that p = Xy + pz . Observe that p | y , so that y = pw for some w ∈ R [ X ]. It follows that p = Xw + z ∈ J . Case II . Suppose that p ∤ R b . Then pR + bR = R , and thus p ∈ pR [ X ] = p R [ X ] + bpR [ X ] ⊆ J . Since X ∈ XJ + pJ , there exist y, z ∈ J such that X = Xy + pz . Note that X | z , and thus z = Xw for some w ∈ R [ X ]. This impliesthat X = y + pw ∈ J .Therefore, either P : J = M or J = M . It then follows that M divides P , andso P = ( P : M ) M . Since M ⊆ P , we have that M ⊆ P : M . Let f ∈ P : M .Then there exist α, β, γ, δ ∈ R [ X ] for which f X = αX + βp and f p = γX + δp .So, f = αX + βX p and f = γp X + δp . Thus, f ∈ ( X, p ) ∩ ( X , p ) = M . Since f was an arbitrary element of P : M , it follows that P : M = M . However, thisimplies that P = ( P : M ) M = M , a contradiction. Therefore, P is a molecule. ✷ We now come to the point where we can present a subclass of Dedekind domains R for which R [ X ] is an FMD. This task is formalized in Corollary 4.9 below as aconsequence to a theorem (Theorem 4.8) revealing how information related to theprime ideals of a domain is enough to guarantee that the domain is an FMD.We first provide a lemma that highlights a particular behavior of invertible idealsthat can be utilized in the context of Noetherian domains. Lemma 4.7.
Let R be a domain that satisfies the ascending chain condition oninvertible ideals and let I be a nonzero ideal of R . Then I is contained in aninvertible ideal of R that is minimal amongst all invertible ideals containing I .Proof. Suppose that I is a nonzero ideal of the domain R which is not containedin an invertible ideal of R that is minimal amongst all invertible ideals contain-ing I . Then there exists a properly descending chain R ) J ) J ) · · · ofinvertible ideals that contain I . Choose some 0 = a ∈ I and put K i = aJ − i for i = 1 , , . . . , so that each K i is necessarily an invertible ideal of R . Moreover, K ( K ( K ( · · · . Therefore, R does not satisfy the ascending chain conditionon invertible ideals, and the result follows. ✷ Theorem 4.8.
Let R be a two-dimensional Noetherian domain such that everyheight-one prime ideal of R is invertible and R/M is finite for every height-twoprime ideal M of R . Then R is an FMD.Proof. First we show that if I is an ideal of R that is not contained in a height-oneprime ideal of R , then I is contained in only finitely many ideals of R .Let I be an ideal of R that is not contained in a height-one prime ideal of R .Observe that if M is a height-two prime ideal of R and n ∈ N , then R/M n is finite(since M k /M k +1 is a finite-dimensional vector space over R/M for each k ∈ W and | R/M n | = Q n − k =0 (cid:12)(cid:12) M k /M k +1 (cid:12)(cid:12) < ∞ ). Let V ( I ) be the set of prime ideals of R containing I . Then V ( I ) consists only of maximal ideals of R . Since R is Noether-ian, V ( I ) is finite, and furthermore the elements of V ( I ) are pairwise comaximal.Hence √ I = Q M ∈ V ( I ) M . Again owing to the fact that R is Noetherian, there issome n ∈ N such that J := Q M ∈ V ( I ) M n ⊆ I . It follows by the Chinese remaindertheorem that R/J ∼ = Q M ∈ V ( I ) R/M n . Consequently, R/J is finite, and thus
R/I is finite. Therefore, I is contained in only finitely many ideals of R . N FINITE MOLECULARIZATION DOMAINS 17
By Corollary 3.5, it is sufficient to show that every nonzero ideal of R is divisibleby only finitely many molecules of R . Let I be a nonzero ideal of R . By Lemma4.7, there is an invertible ideal J of R that contains I and is minimal amongstall of the invertible ideals of R that contain I . Assume that IJ − is containedin a height-one prime ideal P of R . By assumption, P must be invertible. Then I ⊆ P J ⊆ J , and hence P J = J , by minimality of J . However, this implies that P = R , a contradiction. Therefore, IJ − is contained in only finitely many idealsof R by the above work.Now, let K be a molecule of R that divides I . Then I = KA for some ideal A of R . Since R is Noetherian, the set of height-one prime ideals of R that contain I is finite. Thus, it is sufficient to show that K is either a height-one prime ideal of R or IJ − ⊆ K .Suppose that K is contained in a height-one prime ideal P of R . Since P isinvertible by hypothesis, K = P B for some ideal B of R , and hence K = P .On other other hand, suppose K is not contained in a height-one prime ideal of R . Note that there is an invertible ideal B of R that contains J + A and is minimalamongst all the invertible ideals of R that contain J + A . Assume that J = B .Then JB − is a proper invertible ideal of R and thus it is contained in a height-oneprime ideal P of R . But then KAB − = IB − ⊆ JB − ⊆ P . We conclude that AB − ⊆ P , and hence J + A ⊆ BP ⊆ B . But, again by minimality of B , it followsthat BP = B , and so P = R , a contradiction. Therefore, A ⊆ B = J , hence IJ − = KAJ − ⊆ K , as desired. ✷ As in [12] and [22], a domain R is said to have the finite norm property if R/I is finite for every nonzero ideal I of R . Corollary 4.9. (a) If R is a domain such that R [ X ] is an FMD, then R is anFMD.(b) If R is a Dedekind domain with the finite norm property, then R [ X ] is anFMD.Proof. (a) Since R ⊆ R [ X ] is a survival extension, it is an immediate consequenceof Proposition 2.4 and Corollary 3.3 that R has unit-cancellation for ideals. Let I be a nonzero ideal of R . Let I be the set of ideals of R that divide I and J be the set of ideals of R [ X ] that divide I [ X ]. Observe that f : I → J given by f ( L ) = L [ X ] is a well-defined injective map. It follows by Corollary 3.3 that J isfinite, and thus I is finite. We conclude from Corollary 3.3 that R is an FMD.(b) Let K be the quotient field of R . If R = K , then R [ X ] is a PID, and so R [ X ] is an FMD. Thus, we may assume that R = K . As such, it is well-knownthat R [ X ] is a two-dimensional Noetherian domain.We first show that every height-one prime ideal of R [ X ] is invertible. Let P bea height-one prime ideal of R [ X ]. We then have the following two cases: Case I.
Suppose that P ∩ R = 0. Put Q = P ∩ R . Then Q is a nonzero primeideal of R and P = Q [ X ]. Since Q is an invertible ideal of R and P is an extensionideal of Q , we have that P is an invertible ideal of R [ X ]. Case II.
Suppose that P ∩ R = 0. Then there is some nonzero f ∈ R [ X ] forwhich P = f K [ X ] ∩ R [ X ]. Let c ( f ) be the ideal of R generated by the coefficientsof f (that is, c ( f ) is the content of f ). Since R is integrally closed, it follows by [27,Lemme 1] that P = f c ( f ) − [ X ]. Since c ( f ) is an invertible ideal of R , we have that c ( f ) − is an invertible fractional ideal of R . Therefore, c ( f ) − [ X ] is an invertiblefractional ideal of R [ X ], and so P is an invertible ideal of R [ X ].Now, we show that R [ X ] /M is finite for all height-two prime ideals of R [ X ]. Let M be a height-two prime ideal of R [ X ]. Put Q = M ∩ R . Since every upper tozero has height-one, it follows that Q is a nonzero prime ideal of R . In addition,since Q [ X ] ( Q + XR [ X ] and Q [ X ] and Q + XR [ X ] are prime ideals of R [ X ], itfollows that Q [ X ] is a height-one prime ideal of R [ X ]. Moreover, there is a ringisomorphism φ : R [ X ] /Q [ X ] → ( R/Q )[ X ]. Put N = φ ( M/Q [ X ]). Since M is amaximal ideal of R [ X ], it follows that N is a maximal ideal of ( R/Q )[ X ]. However,since R has the finite norm property, R/Q is a finite field, whence (
R/Q )[ X ] hasthe finite norm property. Therefore, R [ X ] /M ∼ = ( R/Q )[ X ] /N is finite, as desired.We conclude that R [ X ] is an FMD by an application of Theorem 4.8. ✷ Corollary 4.10. If O K is the ring of integers of the algebraic number field K , then O K [ X ] is an FMD.Proof. This follows immediately from Corollary 4.9 and the well-known fact thatthe ring of integers in any algebraic number field is a Dedekind domain with thefinite norm property. ✷ FMD’S AND THE D+M CONSTRUCTION
We conclude this paper with a brief section on the intersection of the FMD prop-erty and the classical D + M construction. An exploration of the “ ∗ -factorability”of D + M construction was conducted in [7], where ∗ is a star operation on the frac-tional ideals of the domain. For our purposes here, we limit the discussion to ∗ = d and, inspired by [7, Proposition 7, Corollary 8], first provide a characterization forwhen D + M is a molecular domain. Theorem 5.1.
Let V be a valuation domain of the form V = K + M , where M is the nonzero maximal ideal of V and K is a field. Let D be a proper subring of K , and put R = D + M . Then R is a molecular domain if and only if V is a DVRand D is a field.Proof. First, suppose that R is molecular. Then D is a field by [7, Proposition7]. Now, suppose that V is not a DVR. Then there exists a prime ideal P of V thatis not finitely generated. Thus, P M = P . However, then P is a prime ideal of R for which P = P M in R , as well. This implies that P is a compound prime idealin R , contradicting the fact that prime ideals are molecules in a molecular domain.Therefore, V must be a DVR.Conversely, suppose that V is a DVR and D is a field. Let a ∈ V be such that M = aV . We claim that every nonzero proper ideal of R has the form a n F + a n +1 V for some nonzero D -subspace F of K and n ≥ I be a nonzero proper ideal of R . Since V is a DVR, there exists b ∈ V for which IV = bV and furthermore it must be the case that I = F b + M b in R for some D -subspace F of K [9, Theorem 2.1(n)]. Since every nonzero proper ideal of V isa power of M , one may take b = a n for some n ≥
1. Hence, I = F a n + a n +1 V .Moreover, if F = 0, then I = a n +1 V = a n +1 K + a n +2 V , where K is necessarilynonzero. Thus, F can be assumed to be nonzero, and the claim is established. N FINITE MOLECULARIZATION DOMAINS 19
Now, it is straightforward to verify that for a nonzero proper ideal I of R , it isthe case that I = a n F + a n +1 V = ( aR ) n − ( aF + a V ). One can further show that M ( aF + a V . Thus, aF + a V is a molecule by [20, Proposition 2.2]. Observethat since D is itself a D -subspace of K , then aR must also be a molecule (cf. [7,Example 4]). Thus, ( aR ) n − ( aF + a V ) is a molecularization for I . Therefore, R is a molecular domain. ✷ Our final result, Theorem 5.2, shows that the addition of the simple condition “ K is finite” (where K is the residue field of the valuation domain V ) to the conditionsin Theorem 5.1 above is all that is needed to characterize when D + M is an FMD. Theorem 5.2.
Let V be a valuation domain of the form K + M , where M is thenonzero maximal ideal of V , and K is a field. Let D be a proper subring of K , andput R = D + M . The following are equivalent:(1) R is an FSD;(2) V is a DVR, D is a field, and K is finite;(3) V is a DVR, D is a field, and K ∗ /D ∗ is finite;(4) R is an FFD;(5) R is an FMD.Proof. The equivalence of conditions (1), (2), (3), and (4) was established in[19, Theorem 4.3]. However, Corollary 3.6 shows that in general (1) ⇒ (5) andCorollary 3.4 shows that in general (5) ⇒ (4). The proof is thus complete. ✷ References [1] D.D. Anderson, D.F. Anderson, and M. Zafrullah,
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