Mori Domains Research Articles

Arithmetic of Mori domains and monoids

In this paper we introduce weakly C-monoids as a new class of v-noetherian monoids. Weakly C-monoids generalize C-monoids and make it possible to study multiplicative properties of a wide class of Mori domains, e.g., rings of generalized power series with coefficients in a field and exponents in a finitely generated monoid. The main goal of the paper is to study the question when a weakly C-monoid is locally tame. After having proved a classification theorem for local tameness, we use it to show that every locally tame weakly C-monoid whose complete integral closure has finite class group has finite catenary degree and finite set of distances.

On the Arithmetic of Strongly Primary Monoids

In this paper we study the arithmetic of strongly primary monoids. Numerical monoids and the multiplicative monoids of one-dimensional local Mori domains are main examples of strongly primary monoids. Our investigations focus on local tameness, a basic finiteness property in the theory of non-unique factorizations. It is well-known that locally tame strongly primary monoids have finite catenary degree and finite set of distances.

The w-integral closure of integral domains

Let D be an integral domain with quotient field K, D ¯ the integral closure of D, X an indeterminate over D, and N v = { f ∈ D [ X ] | ( A f ) v = D } . Let w be the ∗-operation on D defined by I w = { x ∈ K | there is a finitely generated ideal A such that A −1 = D and x A ⊆ I } , and let D w = { u ∈ K | u I w ⊆ I w for some nonzero finitely generated ideal I of D } . Then D w , called the w-integral closure of D, is an integrally closed overring of D. In this paper, we show that D w = D ¯ [ X ] N v ∩ K and D w [ X ] N v = D ¯ [ X ] N v . Using this result, we give several w-integral closure analogs of the integral closure. We also study the w-integral closure of UMT-domains and strong Mori domains.

W-Divisorial domains

We study the class of domains in which each w-ideal is divisorial, extending several properties of divisorial and totally divisorial domains to a much wider class of domains. In particular we consider P v MD s and Mori domains.

Semistar Dedekind domains

Let D be an integral domain and ★ a semistar operation on D. As a generalization of the notion of Noetherian domains to the semistar setting, we say that D is a ★-Noetherian domain if it has the ascending chain condition on the set of its quasi-★-ideals. On the other hand, as an extension the notion of Prüfer domain (and of Prüfer v-multiplication domain), we say that D is a Prüfer ★-multiplication domain (P★MD, for short) if D M is a valuation domain, for each quasi-★ f -maximal ideal M of D. Finally, recalling that a Dedekind domain is a Noetherian Prüfer domain, we define a ★-Dedekind domain to be an integral domain which is ★-Noetherian and a P★MD. For the identity semistar operation d, this definition coincides with that of the usual Dedekind domains and when the semistar operation is the v-operation, this notion gives rise to Krull domains. Moreover, Mori domains not strongly Mori are ★-Dedekind for a suitable spectral semistar operation. Examples show that ★-Dedekind domains are not necessarily integrally closed nor one-dimensional, although they mimic various aspects, varying according to the choice of ★, of the “classical” Dedekind domains. In any case, a ★-Dedekind domain is an integral domain D having a Krull overring T (canonically associated to D and ★) such that the semistar operation ★ is essentially “univocally associated” to the v-operation on T. In the present paper, after a preliminary study of ★-Noetherian domains, we investigate the ★-Dedekind domains. We extend to the ★-Dedekind domains the main classical results and several characterizations proven for Dedekind domains. In particular, we obtain a characterization of a ★-Dedekind domain by a property of decomposition of any semistar ideal into a “semistar product” of prime ideals. Moreover, we show that an integral domain D is a ★-Dedekind domain if and only if the Nagata semistar domain Na( D,★) is a Dedekind domain. Several applications of the general results are given for special cases of the semistar operation ★.

Group rings and semigroup rings over strong Mori domains, II

Let R be an integral domain and let S be a torsion-free cancellative additive monoid with quotient group G. We show that the semigroup ring R[ X; S] is a strong Mori domain if and only if R is a strong Mori domain, S is a strong Mori semigroup, and each nonzero element of G is of type (0,0,0,…).

Power series rings over strong Mori domains

We show that: (1) if R is a Noetherian domain, then R〚{X λ} λ∈Λ〛 1 is a strong Mori domain; (2) if R is a strong Mori domain, then R〚{X λ} λ∈Λ〛 1 is a Mori domain; but (3) there exists a strong Mori domain R such that R〚X〛 is not a strong Mori domain.

Injective modules over Mori domains

Injective modules are considered over commutative domains. It is shown that every injective module admits a decomposition into two summands, where one of the summands contains an essential submodule whose elements have divisorial annihilator ideals, while the other summand contains no element with divisorial annihilator. In the special case of Mori domains (i.e., the divisorial ideals satisfy the maximum condition), the first summand is a direct sum of a S-injective module and a module that has no such summand. The former is a direct sum of indecomposable injectives, while the latter is the injective hull of such a direct sum. Those Mori domains R are characterized for which the injective hull of Q/R is S-injective (Q denotes the field of quotients of R) as strong Mori domains, correcting a false claim in the literature.

TW-domains and Strong Mori domains

In this paper we are mainly concerned with TW-domains, i.e., domains in which the w- and t-operations coincide. Precisely, we investigate possible connections with related well-known classes. We characterize the TW-property in terms of divisoriality for Mori domains and Noetherian domains. Specifically, we prove that a Mori domain R is a TW-domain if and only if R M is a divisorial domain for each t-maximal ideal M of R. It turns out that a Mori domain which is a TW-domain is a Strong Mori domain. The last section examines the transfer of the “ TW-domain” and “Strong Mori” properties to pullbacks, in order to provide some original examples.

On overrings of Strong Mori domains

We show that if R is a Strong Mori domain (SM domain) and T is a w-overdomain of R such that T⊆ R wg , then T is an SM domain; if R is an SM domain, then R S is an SM domain for any generalized multiplicative system S of R. Finally, we give an example such that an intersection with finiteness condition of SM domains is not an SM domain.

Group rings and semigroup rings over Strong Mori domains

In this paper we study the transfer of the property of being a Strong Mori domain. In particular we give the characterizations of Strong Mori domains in certain types of pullbacks. We show that if R is a Strong Mori domain which is not a field, then the polynomial ring R[{ X λ } λ∈ Λ ] is also a Strong Mori domain and w- dim R[{X λ} λ∈Λ]=w - dim R . We also determine necessary and sufficient conditions in order that the group ring R[ X; G] or the semigroup ring R[ X; S] should be a Strong Mori domain with w- dimension ≤1.

On Strong Mori domains

In this note we investigate properties of Strong Mori domains, which form a proper subclass of Mori domains. In particular, we show that Strong Mori domains satisfy the Principal Ideal Theorem, the Hubert Basis Theorem and the Krull Intersection Theorem. We also provide some new characterizations of Krull domains and show that the complete integral closure of a Strong Mori domain is a Krull domain.

The Radical Trace Property and Primary Ideals

Let R be an integral domain and let M be an R-module. Then the trace ! Ž . 4 of M is the ideal generated by the set fm fHom M, R and mM . R For an ideal I of R, the trace is simply the product of I and I#1. We call an ideal J a trace ideal if J is the trace of some R-module. An elementary result which will be used freely throughout this paper is that if J is a trace #1 #1 Ž . $ % ideal, then JJ & J; i.e., J equals the ring J : J 6, Proposition 7.2 . Thus the trace ideals of R are precisely those ideals J for which J#1 Ž . Ž equals J : J . Such ideals are also referred to as being ‘‘strong’’; see, for $ % . example, 3 . If R is a valuation domain and M is an R-module, then the $ % trace of M is either R or a prime ideal of R 14, Proposition 2.1 . Extracting the conclusion of this result, Fontana et al. give the following Ž definition: A domain R is said to satisfy the trace property or to be a TP . domain if for each R-module M, the trace of M is equal to either R or a $ % Ž $ %. prime ideal of R 14, p. 169 see also 1, Theorem 2.8 . Theorem 3.5 of $ % 14 gives a characterization of Noetherian TP domains. Namely, for a Noetherian domain R, R is a TP domain if and only if R is onedimensional, has at most one non-invertible maximal ideal M, and, if such #1 Ž a maximal ideal exists, then M equals the integral closure of R or, #1 Ž . . $ % equivalently, M & M : M is a Dedekind domain . In Section 2 of 17 , Gabelli shows that by replacing ‘‘integral closure’’ with ‘‘complete integral closure,’’ the same list of conditions characterizes the class of Mori domains which satisfy the trace property. Recall that a Mori domain is an integral domain which satisfies the ascending chain condition on divisorial ideals.

Sur le groupe des classes d'un anneau intègre

In this paper, we are interested in a generalization of Nagata's theorem [13] to a new class of domains other than Krull domains, the Mori domains and the PVMD domains (Theorem 1).

On polynomial extensions of Mori domains over countable fields

We show here that a polynomial extension of a Mori domain is not necessarily Mori. More precisely, given a field K, there exists a Mori domain A containing K such that A[ X] is not Mori if and only if K is countable. A similar statement holds for the CC ⊥ property.

On Mori domains and commutative rings with CC ⊥ II

We construct here an example of a Mori domain A such that A[[ X]] is not Mori. We show that a finitely generated overring of a factorial domain is not necessarily Mori. We present two criteria for the Mori property. The first criterion leads to a characterization of Mori polynomial extensions. The second criterion is the following: a domain is Mori if and only if it contains a Mori ideal which is prime and unitary or maximal divisorial. We generalize Mori pullbacks.

On Mori domains and commutative rings with CC ⊥ I

We prove here, among other results, that if A is a Mori domain containing an uncountable field, then any polynomial ring over A is Mori. The Mori property of a domain is determined by its maximal ideals. We use extensively a known characterization of Mori domains in terms of the chain condition on annihilators. We characterize Mori domains in terms of semivaluations and divisibility groups. We prove an analogue of Nagata's theorem for the Mori property and present an example of a completely integrally closed domain A, S a multiplicative subset of A generated by prime elements, such that A is completely integrally closed, but A S is not.

Primary decomposition of divisorial ideals in Mori domains

Primary decomposition of divisorial ideals in Mori domains

Power series over mori domains

Let A be a Mori domain; is it true that A[[XJ] is also a Mori domain? I give several instances where the answer is positive, even though I conjecture that there is a Mori domain A such that A[[X1] is not a Mori domain.

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