A note on Itoh (e)-valuation rings of an ideal

2018 ◽  
Vol 498 ◽  
pp. 47-70
Author(s):  
Youngsu Kim ◽  
Louis J. Ratliff ◽  
David E. Rush
Keyword(s):  
Valuation Rings

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 Valuation rings are derived splinters

2021 ◽  
Author(s):  
Benjamin Antieau ◽  
Rankeya Datta
Keyword(s):  
Valuation Rings

scholarly journals Algebraic series and valuation rings over nonclosed fields

2008 ◽  
Vol 212 (8) ◽  
pp. 1996-2010 ◽  
Author(s):  
Steven Dale Cutkosky ◽  
Olga Kashcheyeva
Keyword(s):  
Valuation Rings

Finitely generated modules over valuation rings

1984 ◽  
Vol 12 (15) ◽  
pp. 1795-1812 ◽  
Author(s):  
Luigi Salce ◽  
Paolo Zanardo
Keyword(s):  
Finitely Generated ◽  
Valuation Rings ◽  
Finitely Generated Modules

scholarly journals Valuation rings in finite-dimensional division algebras

1989 ◽  
Vol 120 (1) ◽  
pp. 90-99 ◽  
Author(s):  
H.H Brungs ◽  
Joachim Gräter
Keyword(s):  
Division Algebras ◽  
Finite Dimensional ◽  
Valuation Rings

scholarly journals An extension of Greenberg’s theorem to general valuation rings

2011 ◽  
Vol 139 (1-2) ◽  
pp. 153-166 ◽  
Author(s):  
Laurent Moret-Bailly
Keyword(s):  
Valuation Rings

Some model theory for almost real closed fields

1996 ◽  
Vol 61 (4) ◽  
pp. 1121-1152 ◽  
Author(s):  
Françoise Delon ◽  
Rafel Farré
Keyword(s):  
Model Theory ◽  
Residue Field ◽  
Elementary Equivalence ◽  
First Order ◽  
Valuation Rings ◽  
Closed Fields ◽  
Theory Of Fields ◽  
Model Theory Of Fields ◽  
Elementary Inclusion ◽  
Henselian Valuation

AbstractWe study the model theory of fields k carrying a henselian valuation with real closed residue field. We give a criteria for elementary equivalence and elementary inclusion of such fields involving the value group of a not necessarily definable valuation. This allows us to translate theories of such fields to theories of ordered abelian groups, and we study the properties of this translation. We also characterize the first-order definable convex subgroups of a given ordered abelian group and prove that the definable real valuation rings of k are in correspondence with the definable convex subgroups of the value group of a certain real valuation of k.

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