Evan Houston

UMT-domains and domains with Prüfer integral closure

An integral domain R is said to be a UMT-domain if uppers to zero in R[X) are maximal t-ideals. We show that R is a UMT-domain if and only if its localizations at maximal tdeals have Prufer integral closure. We also prove that the UMT-property is preserved upon passage to polynomial rings. Finally, we characterize the UMT-property in certian pullback constructions; as an application, we show that a domain has Prufer integral closure if and only if all its overrings are UMT-domains.

t-linked extensions, the t-class group, and Nagata's theorem

Abstract Let A be a subring of the integral domain B . Then B is said to be t-linked over A if for each finitely generated ideal I of A with I -1 = A , we have ( IB ) -1 = B . If A and B are Krull domains, this condition is equivalent to PDE. We show that if B is t-linked over A , then the map I →( IB ) t gives a homomorphism from the group of t-invertible t-ideals of A to the group of t-invertible t-ideals of B and hence a homomorphism Cl t ( A )→Cl t ( B ) of the t-class groups. Conditions are given for these maps to be surjective which extend Nagatas Theorem for Krull domains to a much larger class of domains including, e.g., Noetherian domains each of whose grade-one prime ideals has height one.

POWERFUL IDEALS, STRONGLY PRIMARY IDEALS, ALMOST PSEUDO-VALUATION DOMAINS, AND CONDUCIVE DOMAINS

ABSTRACT Let R be a domain with quotient field K, and let I be an ideal of R. We say that I is powerful (strongly primary) if whenever and , we have or ( or for some ). We show that an ideal with either of these properties is comparable to every prime ideal of R, that an ideal is strongly primary it is a primary ideal in some valuation overring of R, and that R admits a powerful ideal R admits a strongly primary ideal R is conducive in the sense of Dobbs-Fedder. Finally, we study domains each of whose prime ideals is strongly primary.

Factoring ideals in integral domains

This volume provides a wide-ranging survey of, and many new results on, various important types of ideal factorization actively investigated by several authors in recent years. Examples of domains studied include (1) those with weak factorization, in which each nonzero, nondivisorial ideal can be factored as the product of its divisorial closure and a product of maximal ideals and (2) those with pseudo-Dedekind factorization, in which each nonzero, noninvertible ideal can be factored as the product of an invertible ideal with a product of pairwise comaximal prime ideals. Prufer domains play a central role in our study, but many non-Prufer examples are considered as well.

Integral Domains Which Admit at Most Two Star Operations

In this article, we characterize domains which admit at most two star operations in the integrally closed and Noetherian cases. We also precisely count the number of star operations on an h-local Prüfer domain.

w-Divisoriality in Polynomial Rings

We extend the Bass–Matlis characterization of local Noetherian divisorial domains to the non-Noetherian case. This result is then used to study the following question: If a domain D is w-divisorial, that is, if each w-ideal of D is divisorial, then is D[X] automatically w-divisorial? We show that the answer is yes if D is either integrally closed or Mori.

Mori domains of integer-valued polynomials

Abstract Let D be a domain with quotient field K . We investigate conditions under which the ring Int (D)={f∈K[X] | f(D)⊆D} of integer-valued polynomials over D is a Mori domain. In particular, we show that if D is a pseudo-valuation domain with finite residue field such that the associated valuation overring is rank one discrete and has infinite residue field, then Int( D ) is a Mori domain with Int( D )≠ D [ X ]. Finally, we investigate the class group of a Mori domain of integer-valued polynomials, showing, in the case just mentioned, that Cl(Int( D )) is generated by the classes of the t -maximal uppers to zero.

Discrete valuation overrings of Noetherian domains

We show that, given a chain 0 = P0 ⊂ P1 ⊂ · · · ⊂ Pn of prime ideals in a Noetherian domain R, there exist a finitely generated overring T of R and a saturated chain of primes in T contracting term by term to the given chain. We further show that there is a discrete rank n valuation overring of R whose primes contract to those of the given chain. Let R be an integral domain (with 1). It is well known that if 0 = P0 ⊂ P1 ⊂ · · · ⊂ Pn is a chain of prime ideals in R, then there exist a valuation overring V of R and a chain of primes 0 = Q0 ⊂ Q1 ⊂ · · · ⊂ Qn in V with Qi ∩ R = Pi for i = 1, . . . , n. On the other hand, Chevalley showed [C] that if R is Noetherian and P is any nonzero prime of R, then there is a discrete rank 1 valuation overring of R centered on P . D. D. Anderson has asked whether these two results can be “combined”: given a Noetherian domain R and a chain as above, is there a discrete rank n valuation overring of R whose primes contract to those of the given chain? (Recall that a finite-dimensional valuation ring V is discrete if PVP is principal for each prime ideal P of V .) We show that this question has an affirmative answer. In fact, we prove the following stronger result. Theorem. Let 0 = P0 ⊂ P1 ⊂ · · · ⊂ Pn be a chain of primes in the Noetherian domain R,and let s1, . . . , sn be a sequence of integers with 1 ≤ si ≤ ht(Pi/Pi−1). Then there exist an overring T of R and a chain of primes 0 = Q0 ⊂ Q1 ⊂ · · · ⊂ Qn in T such that Qi ∩ R = Pi and ht(Qi/Qi−1) = si for i = 1, . . . , n. Moreover, T can be taken to be either a finitely generated extension of R or a discrete valuation ring of rank s = ∑ si. We use “⊂” to denote proper containment. For a prime ideal P of a (commutative) ring R, an upper to P is a prime ideal U of R[X ] for which U ∩ R = P and U 6= P [X ]. An overring of a domain is understood to have the same quotient field as the base domain. Other terminology is standard as in [G]. Lemma 1. Let P be a prime ideal in a Noetherian ring R, and let a, b ∈ R\P . If there is no prime p for which P ⊂ p, ht(p/P ) = 1, and a, b ∈ p, then rad(P, aX − b)R[X ] is an upper to P in R[X ]. Proof. Let U be a prime of R[X ] minimal over (P, aX − b)R[X ]. By the principal ideal theorem, ht(U/PR[X ]) = 1. Hence either U is an upper to P or U = p[X ], Received by the editors October 24, 1994 and, in revised form, December 16, 1994. 1991 Mathematics Subject Classification. Primary 13E05, 13A18; Secondary 13G05, 13A15. c ©1996 American Mathematical Society

On integral domains whose overrings are Kaplansky ideal transforms

Abstract Let R be an integral domain with quotient field K . The Kaplansky transform of an ideal I of R is given by Ω(I)={z ∈ K| rad ((R: R zR))⊇I} . For finitely generated ideals, this agrees with the Nagata transform. We attempt to characterize Ω -domains, that is, domains each of whose overrings is a Kaplansky transform. We obtain a particularly satisfactory characterization when we restrict to the class of Prufer domains: a Prufer domain R is an Ω -domain if and only if for each nonzero branched prime ideal P of R the set P ↓ ={ Q ∈Spec(R)| Q ⊆ P } is open in the Zariski topology.

Toward a classification of prime ideals in Prüfer domains

Abstract The primary purpose of this paper is give a classification scheme for the nonzero primes of a Prüfer domain based on five properties. A prime P of a Prüfer domain R could be sharp or not sharp, antesharp or not, divisorial or not, branched or unbranched, idempotent or not. Based on these five basic properties, there are six types of maximal ideals and twelve types of nonmaximal (nonzero) primes. Both characterizations and examples are given for each type that exists.