We will prove that if the unit rank of a number field with cyclic class group is large enough and if the Galois group of its Hilbert class field over Q is abelian, then every generator of its class group is a Euclidean ideal class. We use this to prove the existence of a non-principal Euclidean ideal class that is not norm-Euclidean by showing that Q( √ 5, √ 21, √ 22) has such an ideal class.

Let R be a (nonzero commutative unital) ring. If I and J are ideals of R such that R/I ⊕R/J is a cyclic R-module, then I + J = R. The rings R such that R/I ⊕R/J is a cyclic R-module for all distinct nonzero proper ideals I and J of R are the following three types of principal ideal rings: fields, rings isomorphic to K ×L for the fields K and L, and special principal ideal rings (R,M) such thatM...

This paper investigates the principal competing visions of global politics that are currently advanced in the public discourse about globalization in opposition to the traditional state-centered perspective. The first part of the paper develops an analysis of ideal models as cultural resources that grounds the different reading of human bonds. The second part applies the notion of ideal models ...

Resultants and Gröbner bases are crucial tools in studying polynomial elimination theory. We investigate relations between the variety of the resultant of two polynomials and the variety of the ideal they generate. Then we focus on the bivariate case, in which the elimination ideal is principal. We study — by means of elementary tools — the difference between the multiplicity of the factors of ...

Let V (resp. D) be a valuation domain (resp. SFT Prüfer domain), I a proper ideal, and V̂ (resp. D̂) be the I-adic completion of V (resp. D). We show that (1) V̂ is a valuation domain, (2) Krull dimension of V̂ = dimV I+1 if I is not idempotent, V̂ ∼= V I if I is idempotent, (3) dim D̂ = dimD I + 1, (4) D̂ is an SFT Prüfer ring, and (5) D̂ is a catenarian ring. Throughout this paper, all rings are assu...

A Weyl arrangement is the arrangement defined by the root system of a finite Weyl group. When a set of positive roots is an ideal in the root poset, we call the corresponding arrangement an ideal subarrangement. Our main theorem asserts that any ideal subarrangement is a free arrangement and that its exponents are given by the dual partition of the height distribution, which was conjectured by ...

This paper investigates the arithmetic of fractional ideals of a purely cubic function field and the infrastructure of the principal ideal class when the field has unit rank one. First, we describe how irreducible polynomials decompose into prime ideals in the maximal order of the field. We go on to compute so-called canonical bases of ideals; such bases are very suitable for computation. We st...