Stefania Gabelli

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.

Ring-theoretic properties of PVMDs

We extend to Prüfer v-multiplication domains some distinguished ring-theoretic properties of Prüfer domains. In particular, we consider the t##-property, the t-radical trace property, w-divisoriality, and w-stability.

Invertible and divisorial ideals of generalized Dedekind domains

Abstract In this paper we give an ideal-theoretical characterization of a distinguished class of Prufer domains, the class of generalized Dedekind domains. Namely, we prove that a Prufer domain R is generalized Dedekind if and only if the divisorial ideals of R are exactly the ideals of type JP l … P n , where J is an invertible fractional ideal and P l , …, P n are (incomparable) nonzero prime ideals of R . We also show that, when R is a generalized Dedekind domain, the group of fractional invertible ideals of R is isomorphic to the free abelian group generated by the set of nonzero prime ideals of R and a basis for it is given by a suitable set of two-generated ideals with prime radical.

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.

On the class group of a Mori domain

It is well known that the monoid D(A) of the divisorial ideals of an integral domain A is a group if and only if A is completely integrally closed. In this case, one may also define the class group of A, C(A) = D(A)/P(A), where P(A) is the subgoup of principal ideals of A. The group C(A) sometimes gives good information about A; for example, if A is a Krull domain, C(A) = 0 if and only if A is a unique factorization domain [9, Proposition 6.11. In [6, 71, the class group C(A) is defined also for a noncompletely integrally closed domain A as C(A) = T(A)/P(A), where 7’(A) is the group of t-invertible t-ideals of A (see the definition in Sect. 1). The aim of this paper is to study the r-invertible t-ideals and the class group of a Mori domain. We recall that a Mori domain is a domain such that the ascending chain condition holds in the set of integral divisorial ideals. Noetherian domains are Mori domains and a Mori domain is a Krull domain if and only if it is completely integrally closed [9, Sect. 33. From this point of view, we begin by observing that in a Mori domain A the t-ideals are exactly the divisorial ideals (Proposition (1.1)); so that T(A) is the group of the invertible elements of D(A), i.e., the u-invertible ideals of A. Since in a Mori domain A a v-invertible divisorial prime is maximal divisorial (Proposition (1.3)), a complete characterization of u-invertible divisorial primes is found early on: they are the maximal divisorial primes P such that A, is a DVR (Corollary (1.4)). Also, we can give a characterization of a certain class of u-invertible divisorial ideals in terms of these primes, generalizing some well-known results for Krull domains

Star Stability and Star Regularity for Mori Domains

In the last few years, the concepts of stability and Clifford regularity have been fruitfully extended by using star operations. In this paper we study and put in relation these properties for Noetherian and Mori domains, substantially improving several results present in the literature.

Complete integral closure and strongly divisorial prime ideals

Abstract It is well known that a domain without proper strongly divisorial ideals is completely integrally closed. In this paper we show that a domain without prime strongly divisorial ideals is not necessarily completely integrally closed, although this property holds under some additional assumptions.

Stability and Clifford-regularity with respect to star operations

In the last few years, the concepts of stability and Clifford regularity have been fruitfully extended by using star operations. In this article we deepen the study of star stable and star regular domains and relate these two classes of domains to each other.

On the class group of integer-valued polynomial rings over Krull domains

Let D be a Krull domain with quotient field K. We study the class group of the integer-valued polynomial ring over D, Int(D)≔{f∈K[X];f(D)⊆D}. In particular, we give necessary and sufficient conditions on D for the class group of Int(D) to be generated by the classes of the t-invertible t-prime ideals and, in this case, we describe its generators. A case of particular interest is when D is a UFD. We also characterize Krull domains D for which Int(D) is a GCD-domain.