Given a stable semistar operation of finite type ⋆ on an integral domain D, we show that it is possible to define in a canonical way a stable semistar operation of finite type [⋆] on the polynomial ring D[X], such that D is a ⋆-quasi-Prüfer domain if and only if each upper to zero in D[X] is a quasi-[⋆]-maximal ideal. This result completes the investigation initiated by Houston-Malik-Mott [18, ...

For certain classes of Prüfer domains A, we study the completion Â,T ofA with respect to the supremum topology T = sup{Tw|w ∈ Ω}, where Ω is the family of nontrivial valuations on the quotient field which are nonnegative on A and Tw is a topology induced by a valuation w ∈ Ω. It is shown that the concepts ‘SFT Prüfer domain’ and ‘generalized Dedekind domain’ are the same. We show that if E is t...

The ring Z consists of the integers of the field Q, and Dedekind takes the theory of unique factorization in Z to be clear and well understood. The problem is that unique factorization can fail when one considers the integers in a finite extension of the rationals, Q(α). Kummer showed that when Q(α) is a cyclotomic extension (i.e. α is a primitive pth root of unity for a prime number p), one ca...

A variety of possible extensions of mappings between posets to their Dedekind order completion is presented. One of such extensions has recently been used for solving large classes of nonlinear systems of partial differential equations with possibly associated initial and/or boundary value problems. 1. The General Setup Let (X,≤) and (Y,≤) be two arbitrary posets and (1.1) φ : X −→ Y any mappin...

The simplest quartic field was introduced by M. Gras and studied by A. J. Lazarus. In this paper, we will evaluate the values of the Dedekind zeta functions at s = −1 of the simplest quartic fields. We first introduce Siegel’s formula for the values of the Dedekind zeta function of a totally real number field at negative odd integers, and will apply Siegel’s formula to the simplest quartic fiel...

Let V (n) denote the number of positive regular integers (mod n) less than or equal to n. We give extremal orders of V (n)σ(n) n2 , V (n)ψ(n) n2 , σ(n) V (n) , ψ(n) V (n) , where σ(n), ψ(n) are the sum-of-divisors function and the Dedekind function, respectively. We also give extremal orders for σ∗(n) V (n) and φ∗(n) V (n) , where σ∗(n) and φ∗(n) represent the sum of the unitary divisors of n a...

The classical class invariants of Weber are introduced as quotients of Thetanullwerte, enabling the computation of these invariants more efficiently than as quotients of values of the Dedekind η-function. We show also how to compute the unit group of suitable ring class fields by means of proving the fact that most of the invariants introduced by Weber are actually units in the corresponding ri...

Let R be a commutative ring, G a group and RG its group ring. Let φ : RG → RG denote the R-linear extension of an involution φ defined on G. An element x in RG is said to be φantisymmetric if φ(x) = −x. A characterization is given of when the φ-antisymmetric elements of RG commute. This is a completion of earlier work. keywords: Involution; group ring; antisymmetric elements. keywords: 2000 Mat...

We show how the usual algorithms valid over Euclidean domains, such as the Hermite Normal Form, the modular Hermite Normal Form and the Smith Normal Form can be extended to Dedekind rings. In a sequel to this paper, we will explain the use of these algorithms for computing in relative extensions of number fields. The goal of this paper is to explain how to generalize to a Dedekind domain R many...