Defining Families for Integral Domains of Real Finite Character

1972 ◽  
Vol 24 (6) ◽  
pp. 1170-1177 ◽  
Author(s):  
William Heinzer ◽  
Jack Ohm
Keyword(s):  
Zero Element ◽  
Finiteness Condition ◽  
Quotient Field ◽  
Integral Domains ◽  
Valuation Rings ◽  
Rank One

Throughout this paper R and D will denote integral domains with the same quotient field K. A set of integral domains {Di} i∊I with quotient field K will be said to have FC (“finite character” or “finiteness condition“) if 0 ≠ ξ ∊ K implies ξ is a unit of Di for all but finitely many i. If ∩i∊IDi also has quotient field K, then {Di} has FC if and only if every non-zero element in ∩i∊IDi is a non-unit in at most finitely many Di. A non-empty set {Vi}i∊:I of rank one valuation rings with quotient field K will be called a defining family of real R-representativesfor D if {Vi} i∊:I has FC, R (⊄ ∩i∊IVi, and D = R∩ (∩i∊I Vi).

Some Open Questions on Minimal Primes of a Krull Domain

1968 ◽  
Vol 20 ◽  
pp. 1261-1264 ◽  
Author(s):  
Paul M. Eakin ◽  
William J. Heinzer
Keyword(s):  
Integral Domain ◽  
Zero Element ◽  
Quotient Field ◽  
Discrete Valuation Rings ◽  
Krull Domain ◽  
Open Questions ◽  
Valuation Rings ◽  
Rank One

Let A be an integral domain and K its quotient field. A is called a Krull domain if there is a set {Vα} of rank one discrete valuation rings such that A = ∩αVα and such that each non-zero element of A is a non-unit in only finitely many of the Vα. The structure of these rings was first investigated by Krull, who called them endliche discrete Hauptordungen (4 or 5, p. 104).

Almost valuation property in bi-amalgamations and pairs of rings

2019 ◽  
Vol 18 (06) ◽  
pp. 1950104 ◽  
Author(s):  
Najib Ouled Azaiez ◽  
Moutu Abdou Salam Moutui
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Quotient Field ◽  
Integral Domains ◽  
Valuation Rings ◽  
Ring Extensions ◽  
Necessary And Sufficient

This paper examines the transfer of the almost valuation property to various constructions of ring extensions such as bi-amalgamations and pairs of rings. Namely, Sec. 2 studies the transfer of this property to bi-amalgamation rings. Our results cover previous known results on duplications and amalgamations, and provide the construction of various new and original examples satisfying this property. Section 3 investigates pairs of integral domains where all intermediate rings are almost valuation rings. As a consequence of our results, we provide necessary and sufficient conditions for a pair (R, T), where R arises from a (T, M, D) construction, to be an almost valuation pair. Furthermore, we study the class of maximal non-almost valuation subrings of their quotient field.

scholarly journals Injective Modules Over Prufer Rings

1959 ◽  
Vol 15 ◽  
pp. 57-69 ◽  
Author(s):  
Eben Matlis
Keyword(s):  
Valuation Ring ◽  
Quotient Field ◽  
Integral Domains ◽  
Injective Modules ◽  
Valuation Rings ◽  
Weakened Form ◽  
Prüfer Rings

The purpose of this paper is to find out what can be learned about valuation rings, and more generally Prufer rings, from a study of their injective modules. The concept of an almost maximal valuation ring can be reformulated as a valuation ring such that the images of its quotient field are injective. The integral domains with this latter property are found to be the Prufer rings with a (possibly) weakened form of linear precompactness for their quotient fields.

scholarly journals Ramification Theory for Extensions of Degree p

1971 ◽  
Vol 41 ◽  
pp. 149-168 ◽  
Author(s):  
Susan Williamson
Keyword(s):  
Residue Class ◽  
Field Extension ◽  
Quotient Field ◽  
Class Field ◽  
Wild Ramification ◽  
Valuation Rings ◽  
Ramification Theory ◽  
Residue Class Field ◽  
Rank One

The notions of tame and wild ramification lead us to make the following definition.Definition. The quotient field extension of an extension of discrete rank one valuation rings is said to be fiercely ramified if the residue class field extension has a nontrivial inseparable part.

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.

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 Crossed Products and Ramification

1966 ◽  
Vol 28 ◽  
pp. 85-111 ◽  
Author(s):  
Susan Williamson
Keyword(s):  
Galois Group ◽  
Galois Extension ◽  
Valuation Ring ◽  
Field Extension ◽  
Integral Closure ◽  
Crossed Product ◽  
Crossed Products ◽  
Quotient Field ◽  
Rank One ◽  
Definition Of

Introduction. Let S be the integral closure of a complete discrete rank one valuation ring R in a finite Galois extension of the quotient field of R, and let G denote the Galois group of the quotient field extension. Auslander and Rim have shown in [3] that the trivial crossed product Δ (1, S, G) is an hereditary order if and only if 5 is a tamely ramified extension of R. And the author has proved in [7] that if the extension S of R is tamely ramified then the crossed product Δ(f, 5, G) is a Π-principal hereditary order for each 2-cocycle f in Z2(G, U(S)). (See Section 1 for the definition of Π-principal hereditary order.) However, the author has exhibited in [8] an example of a crossed product Δ(f, S, G) which is a Π-principal hereditary order in the case when S is a wildly ramified extension of R.

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.

scholarly journals Radical Modules over a Dedekind Domain

1966 ◽  
Vol 27 (2) ◽  
pp. 643-662 ◽  
Author(s):  
A. Fröhlich
Keyword(s):  
Quotient Group ◽  
Multiplicative Group ◽  
Algebraic Closure ◽  
Zero Element ◽  
Dedekind Domain ◽  
Positive Power ◽  
Quotient Field

A radical of a field K is a non zero element of a given algebraic closure some positive power of which lies in K. The group R(K) of radicals reflects properties of the field K and is in turn easily determined as an extension of the multiplicative group K* of non zero elements of K. The elements of the quotient group R(K)/K* are then conveniently identified with certain subspaces of the algebraic closure, the radical spaces of K (cf. §1). What we are here concerned with is the corresponding arithmetic situation, in which we start with a Dedekind domain o with quotient field K. The role of the radicals is taken over by the radical modules. These form a group (o) which contains the group of fractional ideals of o (cf. §4).

Some Properties of Noetherian Domains of Dimension One

1961 ◽  
Vol 13 ◽  
pp. 569-586 ◽  
Author(s):  
Eben Matlis
Keyword(s):  
Vector Space ◽  
Integral Domain ◽  
Chain Condition ◽  
Prime Ideals ◽  
Quotient Field ◽  
Rank One ◽  
Descending Chain Condition ◽  
A Chain ◽  
Dimension One ◽  
Torsion Free

Throughout this discussion R will be an integral domain with quotient field Q and K = Q/R ≠ 0. If A is an R-module, then A is said to be torsion-free (resp. divisible), if for every r ≠ 0 ∈ R the endomorphism of A defined by x → rx, x ∈ A, is a monomorphism (resp. epimorphism). If A is torsion-free, the rank of A is defined to be the dimension over Q of the vector space A ⊗R Q; (we note that a torsion-free R-module of rank one is the same thing as a non-zero R-submodule of Q). A will be said to be indecomposable, if A has no proper, non-zero, direct summands. We shall say that A has D.C.C., if A satisfies the descending chain condition for submodules. By dim R we shall mean the maximal length of a chain of prime ideals in R.

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