On Dedekind domains whose class groups are direct sums of cyclic groups

2023/05/30 by Gyu Whan Chang, Chang, Gyu Whan, Alfred Geroldinger +1
Mathematics · #13A05 #13C20 #13F05 #20M12 #20M25 #Commutative Algebra (math.AC) #FOS: Mathematics #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2305.18796

openalex publication_date 2023/05/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a given family (G_i)_{i ∈ \N} of finitely generated abelian groups, we construct a Dedekind domain D having the following properties. \beginenumerate \item \Pic(D) ≅ \bigoplus_{i ∈ \N}G_i. \item For each i ∈ \N, there exists a submonoid S_i ⊆ D^\bullet with \Pic (D_Si) ≅ G_i. \item Each class of \Pic (D) and of all \Pic (D_Si) contains infinitely many prime ideals. \endenumerate Furthermore, we study orders as well as sets of lengths in the Dedekind domain D and in all its localizations D_Si.

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