DECIDABILITY FOR THEORIES OF MODULES OVER VALUATION DOMAINS

2015 ◽  
Vol 80 (2) ◽  
pp. 684-711 ◽  
Author(s):  
LORNA GREGORY
Keyword(s):  
Residue Field ◽  
Krull Dimension ◽  
Valuation Domain ◽  
Decidable Theory ◽  
Valuation Domains ◽  
Undecidable Theory

AbstractExtending work of Puninski, Puninskaya and Toffalori in [5], we show that if V is an effectively given valuation domain then the theory of all V-modules is decidable if and only if there exists an algorithm which, given a, b ε V, answers whether a ε rad(bV). This was conjectured in [5] for valuation domains with dense value group, where it was proved for valuation domains with dense archimedean value group. The only ingredient missing from [5] to extend the result to valuation domains with dense value group or infinite residue field is an algorithm which decides inclusion for finite unions of Ziegler open sets. We go on to give an example of a valuation domain with infinite Krull dimension, which has decidable theory of modules with respect to one effective presentation and undecidable theory of modules with respect to another. We show that for this to occur infinite Krull dimension is necessary.

Decidability and essential undecidability

1957 ◽  
Vol 22 (1) ◽  
pp. 39-54 ◽  
Author(s):  
Hilary Putnam
Keyword(s):  
Simple Proof ◽  
Arithmetic Functions ◽  
Successor Function ◽  
Open Problems ◽  
Decidable Theory ◽  
Undecidable Theory ◽  
The Common

There are a number of open problems involving the concepts of decidability and essential undecidability. This paper will present solutions to some of these problems. Specifically:(1) Can a decidable theory have an essentially undecidable, axiomatizable extension (with the same constants)?(2) Are all the complete extensions of an undecidable theory ever decidable?We shall show that the answer to both questions is in the affirmative. In answering question (1), the decidable theory for which an essentially undecidable axiomatizable extension will be constructed is the theory of the successor function and a single one-place predicate. It will also be shown that the decidability of this theory is a “best possible” result in the following direction: the theory of either of the common diadic arithmetic functions and a one-place predicate; i.e., of addition and a one-place predicate, or of multiplication and a one-place predicate, is undecidable.Before establishing the main result, it is convenient to give a simple proof that a decidable theory can have an axiomatizable (simply) undecidable extension. This is, of course, an immediate consequence of the main result; but the proof is simple and illustrates the methods that we are going to use in this paper.

On the Weak Global Dimension of Pseudovaluation Domains

1978 ◽  
Vol 21 (2) ◽  
pp. 159-164 ◽  
Author(s):  
David E. Dobbs
Keyword(s):  
Integral Domain ◽  
Global Dimension ◽  
Valuation Domain ◽  
Valuation Domains ◽  
Dimension 2 ◽  
The Ideal

In [7], Hedstrom and Houston introduce a type of quasilocal integral domain, therein dubbed a pseudo-valuation domain (for short, a PVD), which possesses many of the ideal-theoretic properties of valuation domains. For the reader′s convenience and reference purposes, Proposition 2.1 lists some of the ideal-theoretic characterizations of PVD′s given in [7]. As the terminology suggests, any valuation domain is a PVD. Since valuation domains may be characterized as the quasilocal domains of weak global dimension at most 1, a homological study of PVD's seems appropriate. This note initiates such a study by establishing (see Theorem 2.3) that the only possible weak global dimensions of a PVD are 0, 1, 2 and ∞. One upshot (Corollary 3.4) is that a coherent PVD cannot have weak global dimension 2: hence, none of the domains of weak global dimension 2 which appear in [10, Section 5.5] can be a PVD.

scholarly journals On the Factorization of Polynomials Over Discrete Valuation Domains

2014 ◽  
Vol 22 (1) ◽  
pp. 273-280
Author(s):  
Doru Ştefănescu
Keyword(s):  
Valuation Domain ◽  
Discrete Valuation Domain ◽  
Valuation Domains ◽  
Univariate Polynomials ◽  
Factorization Of Polynomials

AbstractWe study some factorization properties for univariate polynomials with coefficients in a discrete valuation domain (A,v). We use some properties of the Newton index of a polynomial to deduce conditions on v(ai) that allow us to find some information on the degree of the factors of F.

scholarly journals Generic formal fibers and analytically ramified stable rings

2013 ◽  
Vol 211 ◽  
pp. 109-135 ◽  
Author(s):  
Bruce Olberding
Keyword(s):  
Prime Ideal ◽  
Noetherian Ring ◽  
Residue Field ◽  
Krull Dimension ◽  
Prime Ideals ◽  
Local Rings ◽  
Embedding Dimension ◽  
Noetherian Domain ◽  
Stable Domains ◽  
Stable Ring

AbstractLet A be a local Noetherian domain of Krull dimension d. Heinzer, Rotthaus, and Sally have shown that if the generic formal fiber of A has dimension d – 1, then A is birationally dominated by a 1-dimensional analytically ramified local Noetherian ring having residue field finite over the residue field of A. We explore further this correspondence between prime ideals in the generic formal fiber and 1-dimensional analytically ramified local rings. Our main focus is on the case where the analytically ramified local rings are stable, and we show that in this case the embedding dimension of the stable ring reflects the embedding dimension of a prime ideal maximal in the generic formal fiber, thus providing a measure of how far the generic formal fiber deviates from regularity. A number of characterizations of analytically ramified local stable domains are also given.

ALGEBRAIC ENTROPY OF ENDOMORPHISMS OVER LOCAL ONE-DIMENSIONAL DOMAINS

2009 ◽  
Vol 08 (06) ◽  
pp. 759-777 ◽  
Author(s):  
PAOLO ZANARDO
Keyword(s):  
Maximal Ideal ◽  
Residue Field ◽  
Field Extension ◽  
Valuation Domain ◽  
One Dimensional ◽  
Natural Properties ◽  
Algebraic Entropy ◽  
Number Of Generators ◽  
Domain V ◽  
Dimensional Integral

Let R be a local one-dimensional integral domain, with maximal ideal 𝔐 and field of fractions Q. Here, a local ring is not necessarily Noetherian. We consider the algebraic entropy ent g, defined using the invariant gen, where, for M a finitely generated R-module, gen (M) is its minimal number of generators. We relate some natural properties of R with the algebraic entropies ent g(ϕ) of the elements ϕ ∈ Q, regarded as endomorphisms in End R(Q). Specifically, let R be dominated by an Archimedean valuation domain V, with maximal ideal P. We examine the uniqueness of V, the transcendency of the residue field extension V/P over R/𝔐, and the condition for R to be a pseudo-valuation domain. We get mutual information between these properties and the behavior of ent g, focusing on the conditions ent g(ϕ) = 0 for every ϕ ∈ Q, ent g(ψ) = ∞ for some ψ ∈ Q, and ent g(ϕ) < ∞ for every ϕ ∈ Q.

scholarly journals Generic formal fibers and analytically ramified stable rings

2013 ◽  
Vol 211 ◽  
pp. 109-135
Author(s):  
Bruce Olberding
Keyword(s):  
Prime Ideal ◽  
Noetherian Ring ◽  
Residue Field ◽  
Krull Dimension ◽  
Prime Ideals ◽  
Local Rings ◽  
Embedding Dimension ◽  
Noetherian Domain ◽  
Stable Domains ◽  
Stable Ring

AbstractLetAbe a local Noetherian domain of Krull dimensiond. Heinzer, Rotthaus, and Sally have shown that if the generic formal fiber ofAhas dimensiond– 1, thenAis birationally dominated by a 1-dimensional analytically ramified local Noetherian ring having residue field finite over the residue field ofA. We explore further this correspondence between prime ideals in the generic formal fiber and 1-dimensional analytically ramified local rings. Our main focus is on the case where the analytically ramified local rings are stable, and we show that in this case the embedding dimension of the stable ring reflects the embedding dimension of a prime ideal maximal in the generic formal fiber, thus providing a measure of how far the generic formal fiber deviates from regularity. A number of characterizations of analytically ramified local stable domains are also given.

scholarly journals Stacked submodules of torsion modules over discrete valuation domains

2003 ◽  
Vol 68 (3) ◽  
pp. 439-447 ◽  
Author(s):  
Pudji Astuti ◽  
Harald K. Wimmer
Keyword(s):  
Valuation Domain ◽  
Torsion Module ◽  
Discrete Valuation Domain ◽  
Valuation Domains ◽  
Torsion Modules

A submodule W of a torsion module M over a discrete valuation domain is called stacked in M if there exists a basis ℬ of M such that multiples of elements of ℬ form a basis of W. We characterise those submodules which are stacked in a pure submodule of M.

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