On Condensed Noetherian Domains Whose Integral Closures are Discrete Valuation Rings

1989 ◽  
Vol 32 (2) ◽  
pp. 166-168 ◽  
Author(s):  
Christian Gottlieb
Keyword(s):  
Valuation Ring ◽  
Integral Domain ◽  
Discrete Valuation Ring ◽  
Discrete Valuation Rings ◽  
Valuation Rings ◽  
Integral Closures

AbstractA condensed domain is an integral domain such that IJ = {xy : x ∊ I, y ∊ J } holds for each pair I, J of ideals. We prove that, under suitable conditions, a subring of a discrete valuation ring is condensed if and only if it contains an element of value 2. We also define the concept strongly condensed.

Polynomial index in discrete valuation rings

2019 ◽  
Vol 56 (2) ◽  
pp. 260-266
Author(s):  
Mohamed E. Charkani ◽  
Abdulaziz Deajim
Keyword(s):  
Lower Bound ◽  
Valuation Ring ◽  
Irreducible Polynomial ◽  
Discrete Valuation Ring ◽  
Field Extension ◽  
Integral Closure ◽  
Quotient Field ◽  
Discrete Valuation Rings ◽  
Valuation Rings ◽  
Adic Valuation

Abstract Let R be a discrete valuation ring, its nonzero prime ideal, P ∈R[X] a monic irreducible polynomial, and K the quotient field of R. We give in this paper a lower bound for the -adic valuation of the index of P over R in terms of the degrees of the monic irreducible factors of the reduction of P modulo . By localization, the same result holds true over Dedekind rings. As an important immediate application, when the lower bound is greater than zero, we conclude that no root of P generates a power basis for the integral closure of R in the field extension of K defined by P.

EXPONENT MATRICES AND TILED ORDERS OVER DISCRETE VALUATION RINGS

2005 ◽  
Vol 15 (05n06) ◽  
pp. 997-1012 ◽  
Author(s):  
V. V. KIRICHENKO ◽  
A. V. ZELENSKY ◽  
V. N. ZHURAVLEV
Keyword(s):  
Markov Chains ◽  
Valuation Ring ◽  
Discrete Valuation Ring ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Discrete Valuation Rings ◽  
Valuation Rings ◽  
Strongly Connected ◽  
Exponent Matrix ◽  
Finite Posets

Exponent matrices appear in the theory of tiled orders over a discrete valuation ring. Many properties of such an order and its quiver are fully determined by its exponent matrix. We prove that an arbitrary strongly connected simply laced quiver with a loop in every vertex is realized as the quiver of a reduced exponent matrix. The relations between exponent matrices and finite posets, Markov chains, and doubly stochastic matrices are discussed.

Witts Theorem for Quadratic Forms Over Non-Dyadic Discrete Valuation Rings

1977 ◽  
Vol 29 (5) ◽  
pp. 928-936
Author(s):  
David Mordecai Cohen
Keyword(s):  
Bilinear Form ◽  
Maximal Ideal ◽  
Quadratic Forms ◽  
Valuation Ring ◽  
Discrete Valuation Ring ◽  
Symmetric Bilinear Form ◽  
Finitely Generated ◽  
Discrete Valuation Rings ◽  
Valuation Rings ◽  
Group Of Isometries

Let R be a discrete valuation ring, with maximal ideal pR, such that ½ ϵ R. Let L be a finitely generated R-module and B : L × L → R a non-degenerate symmetric bilinear form. The module L is called a quadratic module. For notational convenience we shall write xy = B(x, y). Let O(L) be the group of isometries, i.e. all R-linear isomorphisms φ : L → L such that B((φ(x), (φ(y)) = B(x, y).

Discriminant as a product of local discriminants

2017 ◽  
Vol 16 (10) ◽  
pp. 1750198 ◽  
Author(s):  
Anuj Jakhar ◽  
Bablesh Jhorar ◽  
Sudesh K. Khanduja ◽  
Neeraj Sangwan
Keyword(s):  
Maximal Ideal ◽  
Valuation Ring ◽  
Approximation Theorem ◽  
Discrete Valuation Ring ◽  
Integral Closure ◽  
Weak Approximation ◽  
Separable Extension ◽  
Ideal Formula ◽  
Valuation Rings ◽  
Basic Equality

Let [Formula: see text] be a discrete valuation ring with maximal ideal [Formula: see text] and [Formula: see text] be the integral closure of [Formula: see text] in a finite separable extension [Formula: see text] of [Formula: see text]. For a maximal ideal [Formula: see text] of [Formula: see text], let [Formula: see text] denote respectively the valuation rings of the completions of [Formula: see text] with respect to [Formula: see text]. The discriminant satisfies a basic equality which says that [Formula: see text]. In this paper, we extend the above equality on replacing [Formula: see text] by the valuation ring of a Krull valuation of arbitrary rank and completion by henselization. In the course of proof, we prove a generalization of the well-known weak Approximation Theorem which is of independent interest as well.

scholarly journals On the Structure of Affine Flat Group Schemes Over Discrete Valuation Rings, II

2020 ◽  
Author(s):  
Phùng Hô Hai ◽  
João Pedro dos Santos
Keyword(s):  
Discrete Valuation Ring ◽  
Free Module ◽  
Galois Groups ◽  
Affine Group ◽  
Group Schemes ◽  
Infinite Type ◽  
Discrete Valuation Rings ◽  
Valuation Rings ◽  
D Modules ◽  
Tannakian Categories

Abstract In the first part of this work [ 12], we studied affine group schemes over a discrete valuation ring (DVR) by means of Neron blowups. We also showed how to apply these findings to throw light on the group schemes coming from Tannakian categories of $\mathcal{D}$-modules. In the present work, we follow up this theme. We show that a certain class of affine group schemes of “infinite type,” Neron blowups of formal subgroups, are quite typical. We also explain how these group schemes appear naturally in Tannakian categories of $\mathcal{D}$-modules. To conclude, we isolate a Tannakian property of affine group schemes, named prudence, which allows one to verify if the underlying ring of functions is a free module over the base ring. This is then successfully applied to obtain a general result on the structure of differential Galois groups over complete DVRs.

ARITHMETICAL FUNCTIONS IN SEVERAL VARIABLES

2005 ◽  
Vol 01 (03) ◽  
pp. 383-399 ◽  
Author(s):  
EMRE ALKAN ◽  
ALEXANDRU ZAHARESCU ◽  
MOHAMMAD ZAKI
Keyword(s):  
Valuation Ring ◽  
Integral Domain ◽  
Discrete Valuation Ring ◽  
Arithmetical Functions ◽  
Several Variables

In this paper we investigate the ring Ar(R) of arithmetical functions in r variables over an integral domain R. We study a class of absolute values, and a class of derivations on Ar(R). We show that a certain extension of Ar(R) is a discrete valuation ring. We also investigate the metric structure of the ring Ar(R).

2-Prime ideals and their applications

2016 ◽  
Vol 15 (03) ◽  
pp. 1650051 ◽  
Author(s):  
Charef Beddani ◽  
Wahiba Messirdi
Keyword(s):  
Prime Ideal ◽  
Valuation Ring ◽  
Integral Domain ◽  
The Other ◽  
Primary Ideal ◽  
Prime Ideals ◽  
Other Hand ◽  
Valuation Rings ◽  
Integrally Closed

This paper introduces the notion of [Formula: see text]-prime ideals, and uses it to present certain characterization of valuation rings. Precisely, we will prove that an integral domain [Formula: see text] is a valuation ring if and only if every ideal of [Formula: see text] is [Formula: see text]-prime. On the other hand, we will prove that the normalization [Formula: see text] of [Formula: see text] is a valuation ring if and only if the intersection of integrally closed 2-prime ideals of [Formula: see text] is a 2-prime ideal. At the end of this paper, we will give a generalization of some results of Gilmer and Heinzer by studying the properties of domains in which every primary ideal is an integrally closed 2-prime ideal.

Integer-valued skew polynomials

2020 ◽  
pp. 2150114
Author(s):  
Nicholas J. Werner
Keyword(s):  
Valuation Ring ◽  
Integral Domain ◽  
Residue Field ◽  
Discrete Valuation Ring ◽  
Ring Structure ◽  
Polynomial Rings ◽  
Skew Polynomial Ring ◽  
Skew Polynomials ◽  
Skew Polynomial Rings ◽  
Skew Polynomial

For a commutative integral domain [Formula: see text] with field of fractions [Formula: see text], the ring of integer-valued polynomials on [Formula: see text] is [Formula: see text]. In this paper, we extend this construction to skew polynomial rings. Given an automorphism [Formula: see text] of [Formula: see text], the skew polynomial ring [Formula: see text] consists of polynomials with coefficients from [Formula: see text], and with multiplication given by [Formula: see text] for all [Formula: see text]. We define [Formula: see text], which is the set of integer-valued skew polynomials on [Formula: see text]. When [Formula: see text] is not the identity, [Formula: see text] is noncommutative and evaluation behaves differently than it does for ordinary polynomials. Nevertheless, we are able to prove that [Formula: see text] has a ring structure in many cases. We show how to produce elements of [Formula: see text] and investigate its properties regarding localization and Noetherian conditions. Particular attention is paid to the case where [Formula: see text] is a discrete valuation ring with finite residue field.

Primary Ideals in Prüfer Domains

1966 ◽  
Vol 18 ◽  
pp. 1024-1030 ◽  
Author(s):  
Jack Ohm
Keyword(s):  
Prime Ideal ◽  
Valuation Ring ◽  
Integral Domain ◽  
Quotient Ring ◽  
Integral Closure ◽  
Dedekind Domain ◽  
Prüfer Domain ◽  
Valuation Rings ◽  
Dedekind Domains ◽  
Primary Ideals

A Prüfer domain is an integral domainDwith the property that for every proper prime idealPofDthe quotient ringDPis a valuation ring. Examples of such domains are valuation rings and Dedekind domains, a Dedekind domain being merely a noetherian Prüfer domain. The integral closure of the integers in an infinite algebraic extension of the rationals is another example of a Prüfer domain (5, p. 555, Theorem 8). This third example has been studied initially by Krull (4) and then by Nakano (8).

scholarly journals Weighted Homology of Bi-Structures over Certain Discrete Valuation Rings

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 744
Author(s):  
Andrei Bura ◽  
Qijun He ◽  
Christian Reidys
Keyword(s):  
Discrete Valuation Ring ◽  
Homology Theory ◽  
Simplicial Homology ◽  
Discrete Valuation Rings ◽  
Relative Homology ◽  
Valuation Rings ◽  
Contraction Procedure ◽  
Chain Maps ◽  
Homology Module ◽  
Invariant Factors

An RNA bi-structure is a pair of RNA secondary structures that are considered as arc-diagrams. We present a novel weighted homology theory for RNA bi-structures, which was obtained through the intersections of loops. The weighted homology of the intersection complex X features a new boundary operator and is formulated over a discrete valuation ring, R. We establish basic properties of the weighted complex and show how to deform it in order to eliminate any 3-simplices. We connect the simplicial homology, Hi(X), and weighted homology, Hi,R(X), in two ways: first, via chain maps, and second, via the relative homology. We compute H0,R(X) by means of a recursive contraction procedure on a weighted spanning tree and H1,R(X) via an inflation map, by which the simplicial homology of the 1-skeleton allows us to determine the weighted homology H1,R(X). The homology module H2,R(X) is naturally obtained from H2(X) via chain maps. Furthermore, we show that all weighted homology modules Hi,R(X) are trivial for i>2. The invariant factors of our structure theorems, as well as the weighted Whitehead moves facilitating the removal of filled tetrahedra, are given a combinatorial interpretation. The weighted homology of bi-structures augments the simplicial counterpart by introducing novel torsion submodules and preserving the free submodules that appear in the simplicial homology.

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