scholarly journals Algebraic entropy for valuation domains

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
Paolo Zanardo
Keyword(s):  
Addition Theorem ◽  
Valuation Domain ◽  
Algebraic Entropy ◽  
Valuation Domains ◽  
Special Case

AbstractLet R be a non-discrete Archimedean valuation domain, G an R-module, Φ ∈ EndR(G).We compute the algebraic entropy entv(Φ), when Φ is restricted to a cyclic trajectory in G. We derive a special case of the Addition Theorem for entv, that is proved directly, without using the deep results and the difficult techniques of the paper by Salce and Virili [8].

scholarly journals Some properties of the growth and of the algebraic entropy of group endomorphisms

2017 ◽  
Vol 20 (4) ◽  
Author(s):  
Anna Giordano Bruno ◽  
Pablo Spiga
Keyword(s):  
Finite Groups ◽  
Addition Theorem ◽  
Finitely Generated ◽  
Growth Type ◽  
Locally Finite ◽  
Locally Finite Groups ◽  
Algebraic Entropy ◽  
Finitely Generated Groups ◽  
Identity Automorphism ◽  
Inner Automorphisms

AbstractWe study the growth of group endomorphisms, a generalization of the classical notion of growth of finitely generated groups, which is strictly related to algebraic entropy. We prove that the inner automorphisms of a group have the same growth type and the same algebraic entropy as the identity automorphism. Moreover, we show that endomorphisms of locally finite groups cannot have intermediate growth. We also find an example showing that the Addition Theorem for algebraic entropy does not hold for endomorphisms of arbitrary groups.

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.

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 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.

ALMOST PERFECT LOCAL DOMAINS AND THEIR DOMINATING ARCHIMEDEAN VALUATION DOMAINS

2002 ◽  
Vol 01 (04) ◽  
pp. 451-467 ◽  
Author(s):  
PAOLO ZANARDO
Keyword(s):  
Commutative Ring ◽  
Maximal Ideal ◽  
Integral Closure ◽  
Nonzero Ideal ◽  
Valuation Domain ◽  
Local Domain ◽  
Valuation Domains ◽  
Domain V

A commutative ring R is said to be almost perfect if R/I is perfect for every nonzero ideal I of R. We prove that an almost perfect local domain R is dominated by a unique archimedean valuation domain V of its field of quotients Q if and only if the integral closure of R contains an ideal of V. We show how to construct almost perfect local domains dominated by finitely many archimedean valuation domains. We provide several examples illustrating various possible situations. In particular, we construct an almost perfect local domain whose maximal ideal is not almost nilpotent.

Butler Modules Over Valuation Domains

1991 ◽  
Vol 43 (1) ◽  
pp. 48-60 ◽  
Author(s):  
L. Fuchs ◽  
E. Monari-Martinez
Keyword(s):  
Abelian Groups ◽  
Valuation Domain ◽  
Direct Sums ◽  
Valuation Domains ◽  
Commutative Domain ◽  
Torsion Free

Let R be a commutative domain with 1, Q its field of quotients, and M a torsion-free R module. By a balanced submodule of M is meant an RD-submodule N [i.e. rN = N ∩ rM for each r ∈ R] such that, for every R-submodule J of Q, every homomorphism η : J → M/N can be lifted to a homomorphism χ:J → M. This definition extends the notion of balancedness as introduced in abelian groups (see e.g. [10, p. 113]). The balanced-projective R-modules can be characterized as summands of completely decomposable R-modules (i.e. summands of direct sums of submodules of Q). If R is a valuation domain, then such summands are again completely decomposable; see [12, p. 275].

Radical Ideals in Valuation Domains

2007 ◽  
Vol 59 (4) ◽  
pp. 880-896
Author(s):  
John E. van den Berg
Keyword(s):  
Proper Ideal ◽  
Valuation Domain ◽  
Prime Ideals ◽  
Radical Ideal ◽  
Valuation Domains ◽  
Distinguished Ideal ◽  
Radical Ideals

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.

On the projectivity of proper normal curves over valuation domains

2021 ◽  
pp. 2250183
Author(s):  
Hagen Knaf
Keyword(s):  
Projective Dimension ◽  
Valuation Domain ◽  
Zariski Closure ◽  
Noetherian Rings ◽  
Regular Curve ◽  
Generic Fiber ◽  
Discrete Valuation Domain ◽  
Closed Point ◽  
Valuation Domains ◽  
Normal Formula

A theorem of Lichtenbaum states, that every proper, regular curve [Formula: see text] over a discrete valuation domain [Formula: see text] is projective. This theorem is generalized to the case of an arbitrary valuation domain [Formula: see text] using the following notion of regularity for non-noetherian rings introduced by Bertin: the local ring [Formula: see text] of a point [Formula: see text] is called regular, if every finitely generated ideal [Formula: see text] has finite projective dimension. The generalization is a particular case of a projectivity criterion for proper, normal [Formula: see text]-curves: such a curve [Formula: see text] is projective if for every irreducible component [Formula: see text] of its closed fiber [Formula: see text] there exists a closed point [Formula: see text] of the generic fiber of [Formula: see text] such that the Zariski closure [Formula: see text] meets [Formula: see text] and meets [Formula: see text] in regular points only.

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.

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