scholarly journals LOCAL RINGS OF BOUNDED MODULE TYPE ARE ALMOST MAXIMAL VALUATION RINGS

2005 ◽  
Vol 33 (8) ◽  
pp. 2851-2855 ◽  
Author(s):  
François Couchot
Keyword(s):  
Local Rings ◽  
Valuation Rings ◽  
Module Type

scholarly journals Approximating discrete valuation rings by regular local rings

2000 ◽  
Vol 129 (1) ◽  
pp. 37-43
Author(s):  
William Heinzer ◽  
Christel Rotthaus ◽  
Sylvia Wiegand
Keyword(s):  
Local Rings ◽  
Discrete Valuation Rings ◽  
Valuation Rings ◽  
Regular Local Rings

Anneaux de fonctions p-adiques

1995 ◽  
Vol 60 (2) ◽  
pp. 484-497 ◽  
Author(s):  
Luc Bélair
Keyword(s):  
Prime Ideal ◽  
Residue Field ◽  
Local Rings ◽  
First Order ◽  
Valuation Rings ◽  
Integrally Closed ◽  
Definable Functions ◽  
Quotient Rings

AbstractWe study first-order properties of the quotient rings (V)/ by a prime ideal where (V) is the ring of p-adic valued continuous definable functions on some affine p-adic variety V. We show that they are integrally closed Henselian local rings, with a p-adically closed residue field and field of fractions, and they are not valuation rings in general but always satisfy ∀ x, y(x∣y2 ∨ y∣x2).

On Nagata's Result about Height One Maximal Ideals and Depth One Minimal Prime Ideals (I)

2018 ◽  
Vol 85 (3-4) ◽  
pp. 356
Author(s):  
Paula Kemp ◽  
Louis J. Ratliff, Jr. ◽  
Kishor Shah
Keyword(s):  
Integral Closure ◽  
Prime Ideals ◽  
Local Rings ◽  
Finite Sets ◽  
Maximal Ideals ◽  
Valuation Rings ◽  
Canonical Bijection ◽  
Minimal Prime

<p>It is shown that, for all local rings (R,M), there is a canonical bijection between the set <em>DO(R)</em> of depth one minimal prime ideals ω in the completion <em><sup>^</sup>R</em> of <em>R</em> and the set <em>HO(R/Z)</em> of height one maximal ideals <em>̅M'</em> in the integral closure <em>(R/Z)'</em> of <em>R/Z</em>, where <em>Z </em>:<em>= Rad(R)</em>. Moreover, for the finite sets <strong>D</strong> := {<em>V*/V* </em>:<em>= (<sup>^</sup>R/ω)'</em>, ω ∈ DO(R)} and H := {<em>V/V := (R/Z)'<sub><em>̅M'</em></sub>, <em>̅M'</em> ∈ HO(R/Z)</em>}:</p><p>(a) The elements in <strong>D</strong> and <strong>H</strong> are discrete Noetherian valuation rings.</p><p>(b) <strong>D</strong> = {<em><sup>^</sup>V</em> ∈ <strong>H</strong>}.</p>

scholarly journals Truncations of ordered abelian groups

2021 ◽  
Vol 82 (2) ◽  
Author(s):  
Paola D’Aquino ◽  
Jamshid Derakhshan ◽  
Angus Macintyre
Keyword(s):  
Abelian Group ◽  
Model Theory ◽  
Abelian Groups ◽  
Forthcoming Paper ◽  
Peano Arithmetic ◽  
Local Rings ◽  
Principal Ideals ◽  
Valuation Rings ◽  
Nonstandard Models ◽  
Quotient Rings

AbstractWe give axioms for a class of ordered structures, called truncated ordered abelian groups (TOAG’s) carrying an addition. TOAG’s come naturally from ordered abelian groups with a 0 and a $$+$$ + , but the addition of a TOAG is not necessarily even a cancellative semigroup. The main examples are initial segments $$[0, \tau ]$$ [ 0 , τ ] of an ordered abelian group, with a truncation of the addition. We prove that any model of these axioms (i.e. a truncated ordered abelian group) is an initial segment of an ordered abelian group. We define Presburger TOAG’s, and give a criterion for a TOAG to be a Presburger TOAG, and for two Presburger TOAG’s to be elementarily equivalent, proving analogues of classical results on Presburger arithmetic. Their main interest for us comes from the model theory of certain local rings which are quotients of valuation rings valued in a truncation [0, a] of the ordered group $${\mathbb {Z}}$$ Z or more general ordered abelian groups, via a study of these truncations without reference to the ambient ordered abelian group. The results are used essentially in a forthcoming paper (D’Aquino and Macintyre, The model theory of residue rings of models of Peano Arithmetic: The prime power case, 2021, arXiv:2102.00295) in the solution of a problem of Zilber about the logical complexity of quotient rings, by principal ideals, of nonstandard models of Peano arithmetic.

Some self-dual local rings of integers not free over their associated orders

1991 ◽  
Vol 110 (1) ◽  
pp. 5-10 ◽  
Author(s):  
N. P. Byott
Keyword(s):  
Galois Group ◽  
Prime Number ◽  
Group Ring ◽  
Finite Extension ◽  
Free Module ◽  
Abelian Extension ◽  
Local Rings ◽  
Valuation Rings ◽  
Rings Of Integers ◽  
Image Position

Let p be a prime number, and let K be a finite extension of the rational p-adic field ℚp. Let L/K be a finite abelian extension with Galois group G, and let L, K denote the valuation rings of L, K respectively. Then L is a free module of rank 1 over the group algebra KG. Defining the associated order of the extension L/K byL can be viewed as a module over the ring , and a fortiori over the group ring KG.

scholarly journals COMMUTATIVE LOCAL RINGS OF BOUNDED MODULE TYPE

2001 ◽  
Vol 29 (3) ◽  
pp. 1347-1355 ◽  
Author(s):  
F. Couchot
Keyword(s):  
Local Rings ◽  
Module Type

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.

scholarly journals Correction to: Classification of non-local rings with genus two zero-divisor graphs

2021 ◽  
Vol 25 (4) ◽  
pp. 3355-3356
Author(s):  
T. Asir ◽  
K. Mano ◽  
T. Tamizh Chelvam
Keyword(s):  
Zero Divisor ◽  
Local Rings ◽  
Non Local ◽  
Genus Two
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