scholarly journals On three-variable expanders over finite valuation rings

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Le Quang Ham ◽  
Nguyen Van The ◽  
Phuc D. Tran ◽  
Le Anh Vinh
Keyword(s):  
Valuation Ring ◽  
Product Type ◽  
Quadratic Polynomial ◽  
Valuation Rings

AbstractLet {\mathcal{R}} be a finite valuation ring of order {q^{r}}. In this paper, we prove that for any quadratic polynomial {f(x,y,z)\in\mathcal{R}[x,y,z]} that is of the form {axy+R(x)+S(y)+T(z)} for some one-variable polynomials {R,S,T}, we have|f(A,B,C)|\gg\min\biggl{\{}q^{r},\frac{|A||B||C|}{q^{2r-1}}\bigg{\}}for any {A,B,C\subset\mathcal{R}}. We also study the sum-product type problems over finite valuation ring {\mathcal{R}}. More precisely, we show that for any {A\subset\mathcal{R}} with {|A|\gg q^{r-\frac{1}{3}}} then {\max\{|AA|,|A^{d}+A^{d}|\}}, {\max\{|A+A|,|A^{2}+A^{2}|\}}, {\max\{|A-A|,|AA+AA|\}\gg|A|^{\frac{2}{3}}q^{\frac{r}{3}}}, and {|f(A)+A|\gg|A|^{\frac{2}{3}}q^{\frac{r}{3}}} for any one variable quadratic polynomial f.

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.

scholarly journals A BETTER COMPARISON OF - AND -COHOMOLOGIES

2019 ◽  
Vol 236 ◽  
pp. 183-213
Author(s):  
SHANE KELLY
Keyword(s):  
Finite Rank ◽  
Valuation Ring ◽  
Direct Proof ◽  
Integral Closure ◽  
Motivic Cohomology ◽  
Valuation Rings

In order to work with non-Nagata rings which are Nagata “up-to-completely-decomposed-universal-homeomorphism,” specifically finite rank Hensel valuation rings, we introduce the notions of pseudo-integral closure, pseudo-normalization, and pseudo-Hensel valuation ring. We use this notion to give a shorter and more direct proof that $H_{\operatorname{cdh}}^{n}(X,F_{\operatorname{cdh}})=H_{l\operatorname{dh}}^{n}(X,F_{l\operatorname{dh}})$ for homotopy sheaves $F$ of modules over the $\mathbb{Z}_{(l)}$-linear motivic Eilenberg–Maclane spectrum. This comparison is an alternative to the first half of the author’s volume Astérisque 391 whose main theorem is a cdh-descent result for Voevodsky motives. The motivating new insight is really accepting that Voevodsky’s motivic cohomology (with $\mathbb{Z}[\frac{1}{p}]$-coefficients) is invariant not just for nilpotent thickenings, but for all universal homeomorphisms.

scholarly journals On Krull’s Conjecture concerning Valuation Rings

1952 ◽  
Vol 4 ◽  
pp. 29-33 ◽  
Author(s):  
Masayoshi Nagata
Keyword(s):  
Valuation Ring ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Quotient Field ◽  
Counter Example ◽  
Valuation Rings ◽  
Integrally Closed ◽  
Necessary And Sufficient

Previously W. Krull conjectured that every completely integrally closed primary domain of integrity is a valuation ring, The main purpose of the present paper is to construct in §1 a counter example against this conjecture. In § 2 we show a necessary and sufficient condition that a field is a quotient field of a suitable completely integrally closed primary domain of integrity which is not a valuation ring.

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.

Elimination of quantifiers for ordered valuation rings

1987 ◽  
Vol 52 (1) ◽  
pp. 116-128 ◽  
Author(s):  
M. A. Dickmann
Keyword(s):  
Model Theory ◽  
Maximal Ideal ◽  
Valuation Ring ◽  
Quantifier Elimination ◽  
Point Of View ◽  
List Type ◽  
Divisibility Property ◽  
Valuation Rings ◽  
Commutative Domain ◽  
Image Position

Cherlin and Dickmann [2] proved that the theory RCVR of real closed (valuation) rings admits quantifier-elimination (q.e.) in the language ℒ = {+, −, ·, 0, 1, <, ∣} for ordered rings augmented by the divisibility relation “∣”. The purpose of this paper is to prove a form of converse of this result:Theorem. Let T be a theory of ordered commutative domains (which are not fields), formulated in the language ℒ. In addition we assume that:(1) The symbol “∣” is interpreted as the honest divisibility relation: (2) The following divisibility property holds in T:If T admits q.e. in ℒ, then T = RCVR.We do not know at present whether the restriction imposed by condition (2) can be weakened.The divisibility property (DP) has been considered in the context of ordered valued fields; see [4] for example. It also appears in [2], and has been further studied in Becker [1] from the point of view of model theory. Ordered domains in which (DP) holds are called in [1] convexly ordered valuation rings, for reasons which the proposition below makes clear. The following summarizes the basic properties of these rings:Proposition I [2, Lemma 4]. (1) Let A be a linearly ordered commutative domain. The following are equivalent:(a) A is a convexly ordered valuation ring.(b) Every ideal (or, equivalently, principal ideal) is convex in A.(c) A is a valuation ring convex in its field of fractions quot(A).(d) A is a valuation ring and its maximal ideal MA is convex (in A or, equivalently, in quot (A)).(e) A is a valuation ring and its maximal ideal is bounded by ± 1.

Integral Closure of Noetherian Domains and Intersections of Rees Valuation Rings, (II)

2017 ◽  
Vol 84 (1-2) ◽  
pp. 55
Author(s):  
Paula Kemp ◽  
Louis J. Ratliff, Jr. ◽  
Kishor Shah
Keyword(s):  
Valuation Ring ◽  
Disjoint Union ◽  
Integral Closure ◽  
Primary Ideal ◽  
Unique Factorization ◽  
Unique Factorization Domain ◽  
Valuation Rings ◽  
Primary Ideals ◽  
Rees Valuation

<p>Let 1 &lt; s<sub>1</sub> &lt; . . . &lt; s<sub>k</sub> be integers, and assume that κ ≥ 2 (so s<sub>k</sub> ≤ 3). Then there exists a local UFD (Unique Factorization Domain) (R,M) such that:</p><p>(1) Height(M) = s<sub>k</sub>.</p><p>(2) R = R' = ∩{VI (V,N) € V<sub>j</sub>}, where V<sub>j</sub> (j = 1, . . . , κ) is the set of all of the Rees valuation rings (V,N) of the M-primary ideals such that trd((V I N) I (R I M)) = s<sub>j</sub> - 1.</p><p>(3) With V<sub>1</sub>, . . . , V<sub>κ</sub> as in (2), V<sub>1</sub> ∪ . . . V<sub>κ</sub>is a disjoint union of all of the Rees valuation rings of allof the M-primary ideals, and each M-primary ideal has at least one Rees valuation ring in each V<sub>j</sub> .</p>

scholarly journals Commutative rings with comparable regular elements

1996 ◽  
Vol 60 (1) ◽  
pp. 90-108 ◽  
Author(s):  
Paolo Zanardo
Keyword(s):  
Valuation Ring ◽  
Commutative Rings ◽  
The Other ◽  
Regular Elements ◽  
Zero Divisors ◽  
Valuation Rings ◽  
A Chain ◽  
Prüfer Rings ◽  
The Ideal

AbstractLet ℜ be the class of commutative rings R with comparable regular elements, that is, given two non zero-divisors in R, one divides the other. Applying the notion of V-valuation due to Harrison and Vitulli, we define the class V-val of V-valuated rings, which is contained in ℜ and contains the class of Manis valuation rings. We prove that these inclusions of classes are both proper. We investigate Prüfer rings inside ℜ, showing that there exist Prüfer rings which lie in ℜ but not in V-val; we prove that a ring R is a Prüfer valuation ring if and only if it is Prüfer and V-valuated, if and only if its lattice of regular ideals is a chain. Finally, we introduce and investigate the ideal I∞ of a ring R ∈ ℜ, which corresponds to the counterimage of ∞, whenever R is V-valuated.

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.

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