scholarly journals On semi-hereditary orders integral over a commutative valuation ring

2004 ◽  
Vol 276 (2) ◽  
pp. 423-452
Author(s):  
Hidetoshi Marubayashi
Keyword(s):  
Valuation Ring ◽  
Hereditary Orders

scholarly journals II-Principal Hereditary Orders

1968 ◽  
Vol 32 ◽  
pp. 41-65 ◽  
Author(s):  
Susan Williamson
Keyword(s):  
Galois Extension ◽  
Valuation Ring ◽  
Multiplicative Group ◽  
Field Extension ◽  
Integral Closure ◽  
Crossed Product ◽  
Quotient Field ◽  
Group Of Units ◽  
Rank One ◽  
Hereditary Orders

Let S denote the integral closure of a complete discrete rank one valuation ring R in a finite Galois extension of the quotient field of R, G the Galois group of the quotient field extension, and f an element of Z2(G,U(S)) where U(S) denotes the multiplicative group of units of S. A crossed product Δ(f, S, G) whose radical is generated as a left ideal by the prime element II of S is an hereditary order according to the Corollary to Thm. 2. 2 of [2], and we call such a crossed product a II-principal hereditary order. In previous papers the author has studied II-principal hereditary orders Δ(f, S, G) for tamely and wildly ramified extensions S of R (see [10] and [11]). The purpose of this paper is to study II-principal hereditary orders Δ(f, S, G) with no restriction on the extension S of R.

scholarly journals Crossed Products and Hereditary Orders

1963 ◽  
Vol 23 ◽  
pp. 103-120 ◽  
Author(s):  
Susan Williamson
Keyword(s):  
Valuation Ring ◽  
Multiplicative Group ◽  
Field Extension ◽  
Integral Closure ◽  
Crossed Product ◽  
Crossed Products ◽  
Quotient Field ◽  
Group Of Units ◽  
Rank One ◽  
Hereditary Orders

Let S be the integral closure of a discrete rank one valuation ring R in a finite Galois extension of the quotient field of R, and denote the Galois group of the quotient field extension by G. It has been proved by Auslander and Rim in [4] that the trivial crossed product Δ(l, S, G) is an hereditary order for tamely ramified extensions S of R and that Δ(l, S, G) is a maximal order if and only if S is an unramified extension of R. The purpose of this paper is to study the crossed product Δ(f, S, G) where [f] is any element of H2(G, U(S)) and S is a tamely ramified extension of R with multiplicative group of units U(S).

scholarly journals Addendum: Π-principal hereditary orders (Nagoya Math. J. 32 (1968), 41–65)

1974 ◽  
Vol 54 ◽  
pp. 215-216
Author(s):  
Susan Williamson
Keyword(s):  
Direct Product ◽  
Galois Extension ◽  
Valuation Ring ◽  
Residue Class ◽  
Integral Closure ◽  
Quotient Field ◽  
Inertia Group ◽  
Residue Class Field ◽  
Rank One ◽  
Hereditary Orders

Let R denote a complete discrete rank one valuation ring of unequal characteristic, and let p denote the characteristic of the residue class field R̅ of R. Consider the integral closure S of R in a finite Galois extension K of the quotient field k of R. Recall (see Prop. 1.1 of [3]) that the inertia group G0 of K over k is a semi-direct product G0 = J × Gp, where J is a cyclic group of order relatively prime to p and Gp is a normal p-subgroup of G.

scholarly journals Ramification Theory for Extensions of Degree p. II

1972 ◽  
Vol 46 ◽  
pp. 97-109
Author(s):  
Susan Williamson
Keyword(s):  
Valuation Ring ◽  
Quotient Field ◽  
Root Of Unity ◽  
Ramification Index ◽  
Ramification Theory ◽  
Rank One ◽  
The Absolute

Let k denote the quotient field of a complete discrete rank one valuation ring R of unequal characteristic and let p denote the characteristic of R̅; assume that R contains a primitive pth root of unity, so that the absolute ramification index e of R is a multiple of p — 1, and each Gallois extension K ⊃ k of degree p may be obtained by the adjunction of a pth root.

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.

REAL CLOSED RINGS AND ORDERED VALUATION RING

1983 ◽  
Vol 29 (8) ◽  
pp. 417-425 ◽  
Author(s):  
Thomas Becker
Keyword(s):  
Valuation Ring

scholarly journals Barrelled Linearly Topologized Modules Over a Discrete Valuation Ring

2020 ◽  
Vol 12 (1) ◽  
pp. 69
Author(s):  
Dinamérico P. Pombo Jr ◽  
Patricia Couto G. Mauro
Keyword(s):  
Valuation Ring ◽  
Discrete Valuation Ring ◽  
Continuous Linear ◽  
Steinhaus Theorem

In this paper barrelled linearly topologized modules over an arbitrary discrete valuation ring are introduced. A general form of the Banach-Steinhaus theorem for continuous linear mappings on barrelled linearly topologized modules is established and some consequences of it are derived.

Sign in / Sign up

Export Citation Format

Share Document