Abstract
Let R be a non-maximal order in a finite algebraic number field with integral closure R ¯. Although R is not a unique factorization domain, we obtain a positive integer N and a family 𝒬 (called a Cale basis) of primary irreducible elements of R such that x N has a unique factorization into elements of 𝒬 for each x∈R coprime with the conductor of R. Moreover, this property holds for each nonzero x∈R when the natural map Spec(R ¯)→Spec(R) is bijective. This last condition is actually equivalent to several properties linked to almost divisibility properties like inside factorial domains, almost Bézout domains, almost GCD domains.
Highlights
Let K be a number field and OK its ring of integers
Faisant got a unique factorization into a family of irreducibles for any xe where x ∈ R is such that Rx + f = R and e is the exponent of the class group of R [7, Théorème 2]
Q are primary and irreducible and we determine a number N, linked to some integers associated to R, such that xN has a unique factorization into elements of Q for each nonzero x ∈ R
Summary
Volume 10, no 1 (2003), p. 117-131. (http://ambp.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://ambp.cedram.org/legal/). (http://ambp.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://ambp.cedram.org/legal/). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Publication éditée par le laboratoire de mathématiques de l’université Blaise-Pascal, UMR 6620 du CNRS. Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/
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