Valuation Domain Research Articles

Rings Over which the Krull Dimension and the Noetherian Dimension of All Modules Coincide

We denote by 𝒜(R) the class of all Artinian R-modules and by 𝒩(R) the class of all Noetherian R-modules. It is shown that 𝒜(R) ⊆ 𝒩(R) (𝒩(R) ⊆ 𝒜(R)) if and only if 𝒜(R/P) ⊆ 𝒩(R/P) (𝒩(R/P) ⊆ 𝒜(R/P)), for all centrally prime ideals P (i.e., ab ∈ P, a or b in the center of R, then a ∈ P or b ∈ P). Equivalently, if and only if 𝒜(R/P) ⊆ 𝒩(R/P) (𝒩(R/P) ⊆ 𝒜(R/P)) for all normal prime ideals P of R (i.e., ab ∈ P, a, b normalize R, then a ∈ P or b ∈ P). We observe that finitely embedded modules and Artinian modules coincide over Noetherian duo rings. Consequently, 𝒜(R) ⊆ 𝒩(R) implies that 𝒩(R) = 𝒜(R), where R is a duo ring. For a ring R, we prove that 𝒩(R) = 𝒜(R) if and only if the coincidence in the title occurs. Finally, if Q is the quotient field of a discrete valuation domain R, it is shown that Q is the only R-module which is both α-atomic and β-critical for some ordinals α,β ≥ 1 and in fact α = β = 1.

The Bass-Quillen problem on a class of local rings with weak global dimension two

Let (R, m) be a local GCD domain. R is called a U2 ring if there is an element u ∈ m − m2 such that R/(u) is a valuation domain and Ru is a Bezout domain. In this case u is called a normal element of R. In this paper we prove that if R is a U2 ring, then R and R[x] are coherent; moreover, if R has a normal element u and dim(R/(u)) = 1, then every finitely generated projective module over R[X] is free.

Minimal dynamical systems on a discrete valuation domain

We consider isometric dynamical systems on a Legendre set of adiscrete valuation domain with finite residual field. Wecharacterize the minimality of the system by using the structuresequence of the Legendre set and also find the corresponding addingmachine to which such a minimal system is conjugate. The minimalityof affine maps acting on the domain or on the group of units isfully studied.

Sigma-cotorsion modules over valuation domains

We give a characterization of Σ-cotorsion modules over valuation domains in terms of descending chain conditions on certain chains of definable subgroups. We prove that pure submodules, direct products and modules elementarily equivalent to a Σ-cotorsion module are again Σ-cotorsion. Moreover, we describe the structure of Σ-cotorsion modules.

Constituent groups of Clifford semigroups arising from t-closure

The t-class semigroup of an integral domain R, denoted S t ( R ) , is the semigroup of fractional t-ideals modulo its subsemigroup of nonzero principal ideals with the operation induced by ideal t-multiplication. We recently proved that if R is a Krull-type domain [M. Griffin, Rings of Krull type, J. Reine Angew. Math. 229 (1968) 1–27], then S t ( R ) is a Clifford semigroup [S. Kabbaj, A. Mimouni, t-Class semigroups of integral domains, J. Reine Angew. Math. 612 (2007) 213–229]. In this paper, we aim to describe the idempotents of S t ( R ) and the structure of their associated groups. We extend and recover well-known results on class semigroups of valuation domains and Prüfer domains of finite character.

Prime ideals of Bhargava domains

Let D be a domain with quotient field K . For any non-zero x ∈ D , we consider the ring B x ( D ) = { f ∈ K [ X ] ∣ ∀ a ∈ D , f ( x X + a ) ∈ D [ X ] } . These domains are subrings of the ring of integer-valued polynomials Int ( D ) . We study here the prime ideals of B x ( D ) for D a Krull domain. Using some localization properties, we focus first to the case of a discrete valuation domain V , and we determine the prime ideals above m , the maximal ideal of V . In contrast to Int ( V ) where all the primes are of the form M a = { f ∈ Int ( V ) ∣ f ( a ) ∈ m ̂ } , (in one-to-one correspondence with the elements a of V ̂ the m -adic completion of V ) we prove that there are two classes of prime ideals above m in B x ( V ) . We give a complete description of the primes and the maximal ideals of B x ( V ) . By globalization, we obtain the prime ideals of B x ( D ) above a height one prime, for D a Krull domain.

Representations of automorphism groups of finite [formula omitted]-modules of rank two

Let o be a complete discrete valuation domain with finite residue field. In this paper we describe the irreducible representations of the groups Aut ( M ) for any finite o -module M of rank two. The main emphasis is on the interaction between the different groups and their representations. An induction scheme is developed in order to study the whole family of these groups coherently. The results obtained depend on the ring o in a very weak manner, mainly through the degree of the residue field. In particular, a uniform description of the irreducible representations of GL 2 ( o / p ℓ ) is obtained, where p is the maximal ideal of o .

The class semigroup of local one-dimensional domains

Let R be a local one-dimensional domain. We investigate when the class semigroup S ( R ) of R is a Clifford semigroup. We make use of the Archimedean valuation domains which dominate R , as a main tool to study its class semigroup. We prove that if S ( R ) is Clifford, then every element of the integral closure R ̄ of R is quadratic. As a consequence, such an R may be dominated by at most two distinct Archimedean valuation domains, and R ̄ coincides with their intersection. When S ( R ) is Clifford, we find conditions for S ( R ) to be a Boolean semigroup. We derive that R is almost perfect with Boolean class semigroup if, and only if R is stable. We also find results on S ( R ) , through examination of [ V / P : R / M ] and v ( M ) , where V dominates R , and P , M are the respective maximal ideals.

Pseudo-almost valuation domains are quasilocal going-down domains, but not conversely

Ayman Badawi has recently introduced the PAVDs, a class of (commutative integral) domains which is found strictly between the class of APVDs (“almost pseudo valuation domains”) and that of the (necessarily quasilocal) domains having a linearly ordered prime spectrum. It is known that the latter class strictly contains the class of quasilocal going-down domains; it is proved that the class of quasilocal going-down domains strictly contains the class of PAVDs. Consequently, each seminormal PAVD is a divided domain. Moreover, for each n, 1 ≤ n ≤ ∞, an example is constructed of a divided domain (necessarily a quasilocal going-down domain) of Krull dimension n which is not a PAVD. Keywords Pseudo-almost valuation domain, Prime ideal, Going-down domain, Divided domain, Quasilocal, Valuation overring, Root extension, Seminormal, D+M construction, Krull dimension Mathematics Subject Classification (2000) Primary 13B24, 13G05, Secondary 13A15, 13F05

P -adic subsets whose factorials satisfy a generalized Legendre formula

Amice studied the notion of a regular compact subset in a local field K, with valuation v and maximal ideal 𝔐. In her work, she introduced the notion of well distributed sequences and showed that every regular compact subset S admits well distributed sequences and that its factorial sequence (n!S) satisfies a generalized Legendre formula: ν ( n ! s ) = ∑ i = 1 i = ∞ [ n q i ] for every integer n and where qi denotes the number of classes of S modulo 𝔐i. In this article, in more general settings, we show the converse assertions. More precisely, we prove that, for every precompact subset of any discrete valuation domain V, the following assertions are equivalent: the topological closure of S is a regular subset, S admits a very well distributed sequence, S satisfies the generalized Legendre formula.

Irreducibility in the Total Ring of Quotients

Let R be a ring whose total ring of quotients Q is von Neumann regular. We investigate the structure of R when it admits an ideal that is irreducible as a submodule of the total ring of quotients. We characterize those rings which contain a maximal ideal that is irreducible in its total ring of quotients Q. An integral domain has a Q-irreducible ideal which is a maximal ideal if and only if R is a valuation domain. We show that when the total ring of quotients of R is von Neumann regular, then having a maximal ideal that is Q-irreducible is equivalently to certain valuation like properties, including the property that the regular ideals are linearly ordered.

Images and Open Subspaces of SV Spaces

A topological space is finitely an F-space if its Stone–Čech compactification is a union of finitely many closed F-spaces and a space is SV if C(X) has the property that C(X)/P is a valuation domain for each prime ring ideal P of C(X). This article studies the images under open continuous functions and the open subspaces of spaces that are finitely an F-space or are SV. It is shown that an open continuous image of a compact space that is finitely an F-space is finitely an F-space and an open continuous image of certain SV spaces is SV. Also, it is shown cozerosets, but not necessarily open sets, of SV spaces are SV spaces and a similar situation holds for spaces that are finitely an F-space.

Some remarks on Prüfer ⋆-multiplication domains and class groups

Let D be an integral domain with quotient field K and let X be an indeterminate over D. Also, let T : = { T λ | λ ∈ Λ } be a defining family of quotient rings of D and suppose that * is a finite type star operation on D induced by T . We show that D is a P*MD (respectively, P vMD) if and only if ( c D ( f g ) ) * = ( c D ( f ) c D ( g ) ) * (respectively, ( c D ( f g ) ) w = ( c D ( f ) c D ( g ) ) w ) for all 0 ≠ f , g ∈ K [ X ] . A more general version of this result is given in the semistar operation setting. We give a method for recognizing P vMD's which are not P*MD's for a certain finite type star operation *. We study domains D for which the *-class group Cl * ( D ) equals the t-class group Cl t ( D ) for any finite type star operation *, and we indicate examples of P vMD's D such that Cl * ( D ) ⊊ Cl t ( D ) . We also compute Cl v ( D ) for certain valuation domains D.

ON MAXIMAL NON-ACCP SUBRINGS

A domain R is a maximal non-ACCP subring of its quotient field if and only if R is either a two-dimensional valuation domain with a DVR overring or a one-dimensional nondiscrete valuation domain. If R ⊂ S is a minimal ring extension and S is a domain, then (R,S) is a residually algebraic pair. If S is a domain but not a field, a maximal non-ACCP subring extension R ⊂ S is a minimal ring extension if (R,S) is a residually algebraic pair and R is quasilocal. Results with a similar flavor are given for domains R ⊂ S sharing a nonzero ideal, with applications to rings R of the form A + XB[X] or A + XB[[X]]. If R ⊂ S is a minimal ring extension such that R is a domain and S is not (R-algebra isomorphic to) an overring of R, then R satisfies ACCP if and only if S satisfies ACCP.

Valuation Domains Whose Products of Free Modules Are Separable

It is proved that if R is a valuation domain with maximal ideal P and if R L is countably generated for each prime ideal L, then R R is separable if and only R J is maximal, where J = ∩ n∈ℕ P n .

Radical Ideals in Valuation Domains

AbstractAn idealIof a ringRis called a radical ideal ifI= ℛ(R) where ℛ is a radical in the sense of Kurosh–Amitsur. The main theorem of this paper asserts that ifRis a valuation domain, then a proper idealIofRis a radical ideal if and only ifIis a distinguished ideal ofR(the latter property means that ifJandKare ideals ofRsuch thatJ⊂I⊂Kthen we cannot haveI/J≅K/Ias rings) and that such an ideal is necessarily prime. Examples are exhibited which show that, unlike prime ideals, distinguished ideals are not characterizable in terms of a property of the underlying value group of the valuation domain.

Rings of Continuous Functions on Spaces of Finite Rank and the SV Property

Let X be a compact topological space and let C(X) denote the f-ring of all continuous real-valued functions defined on X. A point x in X is said to have rank n if, in C(X), there are n minimal prime ℓ-ideals contained in the maximal ℓ-ideal M x = {f ∈ C(X):f(x) = 0}. The space X has finite rank if there is an n ∈ N such that every point x ∈ X has rank at most n. We call X an SV space (for survaluation space) if C(X)/P is a valuation domain for each minimal prime ideal P of C(X). Every compact SV space has finite rank. For a bounded continuous function h defined on a cozeroset U of X, we say there is an h-rift at the point z if h cannot be extended continuously to U ∪ {z}. We use sets of points with h-rift to investigate spaces of finite rank and SV spaces. We show that the set of points with h-rift is a subset of the set of points of rank greater than 1 and that whether or not a compact space of finite rank is SV depends on a characteristic of the closure of the set of points with h-rift for each such h. If X has finite rank and the set of points with h-rift is an F-space for each h, then X is an SV space. Moreover, if every x ∈ X has rank at most 2, then X is an SV space if and only if for each h, the set of points with h-rift is an F-space.

On finitely injective modules and locally pure-injective modules over Prüfer domains

Over Matlis valuation domains there exist finitely injective modules which are not direct sums of injective modules, as well as complete locally pure-injective modules which are not the completion of a direct sum of pure-injective modules. Over Prufer domains which are either almost maximal, or h-local Matlis, finitely injective torsion modules and complete torsion-free locally pure-injective modules correspond to each other under the Matlis equivalence. Almost maximal Priifer domains are characterized by the property that every torsion-free complete module is locally pure-injective. It is derived that semi-Dedekind domains are Dedekind.

On 2-absorbing ideals of commutative rings

Suppose that R is a commutative ring with 1 ≠ 0. In this paper, we introduce the concept of 2-absorbing ideal which is a generalisation of prime ideal. A nonzero proper ideal I of R is called a 2-absorbing ideal of R if whenever a, b, c ∈ R and abc ∈ I, then ab ∈ I or ac ∈ I or bc ∈ I. It is shown that a nonzero proper ideal I of R is a 2-absorbing ideal if and only if whenever I1I2I3 ⊆ I for some ideals I1,I2,I3 of R, then I1I2 ⊆ I or I2I3 ⊆ I or I1I3 ⊆ I. It is shown that if I is a 2-absorbing ideal of R, then either Rad(I) is a prime ideal of R or Rad(I) = P1 ⋂ P2 where P1,P2 are the only distinct prime ideals of R that are minimal over I. Rings with the property that every nonzero proper ideal is a 2-absorbing ideal are characterised. All 2-absorbing ideals of valuation domains and Prüfer domains are completely described. It is shown that a Noetherian domain R is a Dedekind domain if and only if a 2-absorbing ideal of R is either a maximal ideal of R or M2 for some maximal ideal M of R or M1M2 where M1,M2 are some maximal ideals of R. If RM is Noetherian for each maximal ideal M of R, then it is shown that an integral domain R is an almost Dedekind domain if and only if a 2-absorbing ideal of R is either a maximal ideal of R or M2 for some maximal ideal M of R or M1M2 where M1,M2 are some maximal ideals of R.

On Pseudo-Almost Valuation Domains

Let R be an integral domain with quotient field K and integral closure R ′. Anderson and Zafrullah called R an “almost valuation domain” if for every nonzero x ∈ K, there is a positive integer n such that either x n ∈ R or x −n ∈ R. In this article, we introduce a new closely related class of integral domains. We define a prime ideal P of R to be a “pseudo-strongly prime ideal” if, whenever x, y ∈ K and xyP ⊆ P, then there is a positive integer m ≥ 1 such that either x m ∈ R or y m P ⊆ P. If each prime ideal of R is a pseudo-strongly prime ideal, then R is called a “pseudo-almost valuation domain” (PAVD). We show that the class of valuation domains, the class of pseudo-valuation domains, the class of almost valuation domains, and the class of almost pseudo-valuation domains are properly contained in the class of pseudo-almost valuation domains; also we show that the class of pseudo-almost valuation domains is properly contained in the class of quasilocal domains with linearly ordered prime ideals. Among the properties of PAVDs, we show that an integral domain R is a PAVD if and only if for every nonzero x ∈ K, there is a positive integer n ≥ 1 such that either x n ∈ R or ax −n ∈ R for every nonunit a ∈ R. We show that pseudo-almost valuation domains are precisely the pullbacks of almost valuation domains, we characterize pseudo-almost valuation domains of the form D + M, and we use this characterization to construct PAVDs that are not almost valuation domains. We show that if R is a Noetherian PAVD, then R has Krull dimension at most one and R ′ is a valuation domain; we show that every overring of a PAVD R is a PAVD iff R ′ is a valuation domain and every integral overring of R is a PAVD.