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).

Convex Directed Subgroups of a Group of Divisibility

1974 ◽  
Vol 26 (3) ◽  
pp. 532-542 ◽  
Author(s):  
Joe L. Mott
Keyword(s):  
Integral Domain ◽  
Multiplicative Group ◽  
Ordered Group ◽  
Quotient Field ◽  
Partially Ordered ◽  
Group Of Units ◽  
Partially Ordered Group ◽  
Group Of Divisibility

If D is an integral domain with quotient field K, the group of divisibility G(D) of D is the partially ordered group of non-zero principal fractional ideals with aD ≦ bD if and only if aD contains bD. If K* denotes the multiplicative group of K and U(D) the group of units of D, then G(D) is order isomorphic to K*/U(D), where aU(D) ≦ bU(D) if and only if b/a ∊ D.

C-Valuations and Normal C-Orderings

1989 ◽  
Vol 41 (1) ◽  
pp. 14-67 ◽  
Author(s):  
M. Chacron
Keyword(s):  
Abelian Group ◽  
Normal Subgroup ◽  
Division Ring ◽  
Maximal Ideal ◽  
Quotient Group ◽  
Valuation Ring ◽  
Multiplicative Group ◽  
Invertible Elements ◽  
Natural Way

Let D stand for a division ring (or skewfield), let G stand for an ordered abelian group with positive infinity adjoined, and let ω: D → G. We call to a valuation of D with value group G, if ω is an onto mapping from D to G such that(i) ω(x) = ∞ if and only if x = 0,(ii) ω(x1 + x2) = min(ω (x1), ω (x2)), and(iii) ω (x1 x2) = ω (x1) + ω (x2).Associated to the valuation ω are its valuation ringR = ﹛x ∈ Dω(x) ≧ 0﹜,its maximal idealJ = ﹛x ∈ |ω(x) > 0﹜, and its residue division ring D = R/J.The invertible elements of the ring R are called valuation units. Clearly R and, hence, J are preserved under conjugation so that 1 + J is also preserved under conjugation. The latter is thus a normal subgroup of the multiplicative group Dm of D and hence, the quotient group D˙/1 + J makes sense (the residue group of ω). It enlarges in a natural way the residue division ring D (0 excluded, and addition “forgotten“).

scholarly journals Effective results for unit points on curves over finitely generated domains

2015 ◽  
Vol 158 (2) ◽  
pp. 331-353
Author(s):  
ATTILA BÉRCZES
Keyword(s):  
Algebraic Closure ◽  
Unit Group ◽  
Number Fields ◽  
Finitely Generated ◽  
Quotient Field ◽  
Effective Version ◽  
Formula Group ◽  
Commutative Domain ◽  
Image Position ◽  
Unit Equations

AbstractLet A be a commutative domain of characteristic 0 which is finitely generated over ℤ as a ℤ-algebra. Denote by A* the unit group of A and by K the algebraic closure of the quotient field K of A. We shall prove effective finiteness results for the elements of the set \begin{equation*} \mathcal{C}:=\{ (x,y)\in (A^*)^2 | F(x,y)=0 \} \end{equation*} where F(X, Y) is a non-constant polynomial with coefficients in A which is not divisible over K by any polynomial of the form XmYn - α or Xm - α Yn, with m, n ∈ ℤ⩾0, max(m, n) > 0, α ∈ K*. This result is a common generalisation of effective results of Evertse and Győry [12] on S-unit equations over finitely generated domains, of Bombieri and Gubler [5] on the equation F(x, y) = 0 over S-units of number fields, and it is an effective version of Lang's general but ineffective theorem [20] on this equation over finitely generated domains. The conditions that A is finitely generated and F is not divisible by any polynomial of the above type are essentially necessary.

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).

scholarly journals Integral ∨-ideals

1981 ◽  
Vol 22 (2) ◽  
pp. 167-172 ◽  
Author(s):  
David F. Anderson
Keyword(s):  
Maximal Ideal ◽  
Integral Domain ◽  
Dedekind Domain ◽  
Integral Ideal ◽  
Quotient Field ◽  
Fractional Ideal

Let R be an integral domain with quotient field K. A fractional ideal I of R is a ∨-ideal if I is the intersection of all the principal fractional ideals of R which contain I. If I is an integral ∨-ideal, at first one is tempted to think that I is actually just the intersection of the principal integral ideals which contain I.However, this is not true. For example, if R is a Dedekind domain, then all integral ideals are ∨-ideals. Thus a maximal ideal of R is an intersection of principal integral ideals if and only if it is actually principal. Hence, if R is a Dedekind domain, each integral ∨-ideal is an intersection of principal integral ideals precisely when R is a PID.

scholarly journals INTEGRAL DOMAINS WHOSE SIMPLE OVERRINGS ARE INTERSECTIONS OF LOCALIZATIONS

2005 ◽  
Vol 04 (02) ◽  
pp. 195-209 ◽  
Author(s):  
MARCO FONTANA ◽  
EVAN HOUSTON ◽  
THOMAS LUCAS
Keyword(s):  
Dedekind Domain ◽  
Quotient Field ◽  
Integral Domains ◽  
Finiteness Conditions ◽  
Integrally Closed ◽  
Prufer Domains ◽  
Prüfer Domains

Call a domain R an sQQR-domain if each simple overring of R, i.e., each ring of the form R[u] with u in the quotient field of R, is an intersection of localizations of R. We characterize Prüfer domains as integrally closed sQQR-domains. In the presence of certain finiteness conditions, we show that the sQQR-property is very strong; for instance, a Mori sQQR-domain must be a Dedekind domain. We also show how to construct sQQR-domains which have (non-simple) overrings which are not intersections of localizations.

The Positive Power of the Reviewing Process

2009 ◽  
Vol 51 (2) ◽  
pp. 1-4
Author(s):  
Agnes Nairn
Keyword(s):  
Journal Articles ◽  
Positive Power ◽  
Peer Reviewing ◽  
Review Procedure

Agnes Nairn shares some thoughts on the peer reviewing process of journal articles, including the role of the reviewer and review procedure.

scholarly journals Méthodes fonctionnelles pour la transcendance en caractéristique finie

1994 ◽  
Vol 50 (2) ◽  
pp. 273-286 ◽  
Author(s):  
Laurent Denis
Keyword(s):  
Exponential Function ◽  
Characteristic Zero ◽  
Algebraic Closure ◽  
Algebraic Groups ◽  
Zero Element ◽  
Finite Characteristic ◽  
First Derivative ◽  
Linearly Independent ◽  
Divisibility Properties

There are essentially two ways to obtain transcendence results in finite characteristic. The first, historically, is to use Ore's lemma and to prove that a series whose coefficients satisfy well-behaved divisibility properties cannot be a zero of an additive polynomial. This method is of the same kind as the method of p–automata. The second one is to try to imitate the usual methods in characteristic zero and to do transcendence theory with t–modules analogously to what we can do with algebraic groups. We want to show here that transcendence results over Fq(T) can also be obtained with the help of the variable T. If ec(z) is the Carlitz exponential function and e = ec(1), we obtain, in particular, that 1, e, …, e(p–2) (the P–2 first derivative of e with respect to T) are linearly independent over the algebraic closure of Fq(T). A corollary is that for every non-zero element α in Fq((1/T)), αpe and αec(e1/p) are transcendental over Fq(T). By changing the variable and using older results we also obtain the transcendence of ec(ω) for all ω ∈ Fq((1/T)) such that ω(T) and ω(Ti) are not zero and linearly dependent over Fq (Ti) (q > 2i + 1). Such u appear to be transcendental by the method of Mahler if i is not a power of p.

An Epi-Reflector for Universal Theories

1973 ◽  
Vol 16 (2) ◽  
pp. 167-171 ◽  
Author(s):  
Paul D. Bacsich
Keyword(s):  
Algebraic Closure ◽  
Amalgamation Property ◽  
Injective Hull ◽  
Universal Theory ◽  
Quotient Field ◽  
Injective Hulls

A construction of an epi-reflector by injective hull techniques is given which applies to the class of models of any universal theory with the Amalgamation Property and there yields a weak but functorial type of algebraic closure. Various completions such as the boolean envelope and quotient field constructions are identified as such injective hulls over epimorphic injections. Forms of the Amalgamation Property are also shown to eliminate various pathologies of epimorphisms and equalizers.

The Positive Power of the Reviewing Process

2009 ◽  
Vol 51 (2) ◽  
pp. 1-4 ◽  
Author(s):  
Agnes Nairn
Keyword(s):  
Journal Articles ◽  
Positive Power ◽  
Peer Reviewing ◽  
Review Procedure

Agnes Nairn shares some thoughts on the peer reviewing process of journal articles, including the role of the reviewer and review procedure.

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