Valentina Barucci

Strongly divisorial ideals and complete integral closure of an integral domain

Throughout this paper, D will be a commutative integral domain with identity and K will denote its quotient field. Moreover, if I# (0) and J are fractional ideals of D, we denote the fractional ideal of D (J;cZ) = {x E K 1 XZC J} simply by (.Z: I), and, as usual, (D i I) by I‘, and (I-‘) ~ ’ by Z,. An ideal Z is divisiorial if Z = Z, . In the first part we connect the concept of strongly divisiorial ideal (cf. [lo, p. 3441) with some recent papers by J. Huckaba, I. Papick (cf. [S]) and D. F. Anderson (cf. Cl]), also introducing the natural concept of “strong” ideal. Proposition 6 shows that the strongly divisorial ideals of D are the nonzero “conductors” of D in some overring of D and hence we easily deduce (cf. Corollary 8) the existence of a one-to-one correspondence between the set of strongly divisorial ideals of D and the set of overrings of D of the type I-‘, i.e., duals of some ideal Z of D. There is a strict relationship between the set of strongly divisorial ideals of D, D,(D), and the complete integral closure D* of D. In fact it turns out that D* = U {I-‘IZED,(D)} (cf. Proposition 12) and D is completely integrally closed if and only if D itself is the unique strongly divisiorial ideal of D. Actually, at each step, passing from D to an overring of the type I-‘, the set of strongly divisorial ideals gets smaller (cf. Proposition 10). Two complementary situations concerning the complete integral closure of a domain are characterized in terms of their strongly divisorial ideals. In the first case, when (D: D*) # (0) (i.e., D is “nomal,” a terminology introduced by J. QuerrC [lo]) the complete integral clousure of D is exactly the dual of a (minimum) strongly divisorial ideal Z of D (cf.

Analytically unramified one-dimensional semilocal rings and their value semigroups

Abstract In a one-dimensional local ring R with finite integral closure each nonzerodivisor has a value in N d , where d is the number of maximal ideals in the integral closure. The set of values constitutes a semigroup, the value semigroup of R. We investigate the connection between the value semigroup and the ring. There is a particularly close connection for some classes of rings, e.g. Gorenstein rings, Arf rings, and rings of small multiplicity. In many respects, the Arf rings and the Gorenstein rings turn out to be opposite extremes. We give applications to overrings, intersection numbers, and multiplicity sequences in the blow-up sequences studied by Lipman.

A Family of Quotients of the Rees Algebra

A family of quotient rings of the Rees algebra associated to a commutative ring is studied. This family generalizes both the classical concept of idealization by Nagata and a more recent concept, the amalgamated duplication of a ring. It is shown that several properties of the rings of this family do not depend on the particular member.

CONDUCIVE INTEGRAL DOMAINS AS PULLBACKS

This article presents a characterization of conducive (integral) domains as the pullbacks in certain types of Cartesian squares in the category of commutative rings. Such squares behave well with respect to (semi)normalization, thus permitting us to recover the recent characterization by Dobbs-Fedder of seminormal conducive domains. The conducive domains satisfying various finiteness conditions (Noetherian, Archimedean, accp) are characterized by identifying suitable restrictions on the data in the corresponding Cartesian squares. Various necessary or sufficient conditions are given for Mori conducive domains. Consequently, one has examples of several (accp non-Mori; Mori non-Noetherian) conducive domains.

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

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.

Gorenstein conducive domains

Communications in Algebra Publication details, including instructions for authors and subscription information: http://www.informaworld.com/smpp/title~content=t713597239 Gorenstein conducive domains David E. Dobbs a; Valentina Barucci b; Marco Fontana b a Department of Mathematics, University ofTennessee, Knoxville, TN, USA b Dipartimento di Matematica, Istituto Guido Casteinuovo, Univerist di Roma La Sapienza, Italia, Roma

Integrally closed factor domains

VALENTINA BARUCCI, DAVID E. DOBBS AND S.B. MULAYThis paper characterises the integral domains R with the property that R/P is integrallyclosed for each prime ideal P of R. It is shown that Dedekind domains are the onlyNoetherian domains with this property. On the other hand, each integrally closed going-down domain has this property. Related properties and examples are also studied.1 INTRODUCTION

When areD+M rings laskerian?

Si studia il passaggio della proprietà di laskerianità in alcuni tipi di anelli ottenuti per prodotto fibrato, estendendo alcuni risultati recenti di I. Armeanu. Si caratterizzano gli anelli di pseudo-valutazione che sono anelli (fortemente) laskeriani. Si costruiscono varie classi di anelli (fortemente) laskeriani non noetheriani.

On the biggest maximally generated ideal as the conductor in the blowing up ring

Let (R, M) be a Noetherian one-dimensional local ring.C Gottlieb calls anM-primary idealI maximally generated ifμ(I)=ℓ(R/(r)), or which is the same, ifIM=rI for somer∈M, and he also proves that if there is a maximally generated ideal inR then there is a unique biggest one (see [4]). In this paper each ring (R, M) is a local one-dimensional Cohen-Macaulay ring. LetQ be the total ring of fractions ofR, and letB(M) be the ring obtained by blowing upM, i.e.B (M)=Ui≥1 (Mi:Mi)Q. We prove in Theorem 1 that if there are maximally generated ideals inR then they are theM-primary ideals ofR which are ideals ofB(M) too. And the biggest maximally generated idealÎ ofR is the conductor ofR inB(M), i.e.(R∶B(M))R. We give in Theorem 3 an algorithm for findingÎ when the integral closure ofR is a local domain with the same residue field asR. In section 3 there are applications to semigroup rings. We prove thatÎ is generated by monomials in Proposition 7, and therefore semigroups are considered in the continuation. Let σ be the reduction exponent ofM, i.e. δ=min{i∶ℓ(Mi/Mi+1) =e(M)} wheree(M) denotes the multiplicity ofM. In Proposition 10, δ is determined, and there is also given a sufficient condition forÎ not to be a power ofM. In Propositions 11 and 12Î is determined for two special cases of semigroup rings whereÎ is a power ofM.