Arithmetic of seminormal weakly Krull monoids and domains

Abstract

We study the arithmetic of seminormal v-noetherian weakly Krull monoids with nontrivial conductor which have finite class group and prime divisors in all classes. These monoids include seminormal orders in holomorphy rings in global fields. The crucial property of seminormality allows us to give precise arithmetical results analogous to the well-known results for Krull monoids having finite class group and prime divisors in each class. This allows us to show, for example, that unions of sets of lengths are intervals and to provide a characterization of half-factoriality.2010 Mathematics Subject Classification. 13A05, 13F05, 13F15, 13F45, 20M13.

A characterization of seminormal C-monoids

“…We mention two results valid for special classes of C-domains. There is a characterization of half-factoriality for orders in quadratic number fields in number theoretic terms ([10, Theorem 3.7.15]) and for a class of seminormal weakly Krull domains (including seminormal orders in number fields) in algebraic terms involving the v-class group and extension properties of prime divisorial ideals (see [11,Theorem 6.2] and [13]).…”

“…Let k ∈ [0, n]. By [10, Proposition 2.9.9], H k ⊂ F k is a C-monoid and it is a divisor-closed submonoid of H. Since H k is a divisor-closed submonoid of the seminormal monoid H, it is seminormal (this is easy to check; for details see [11,Lemma 3.2]). By [10, Lemma 2.8.4.5], ψ k :…”

Products of k atoms in Krull monoids

“…Their suprema ρ k (H) = sup U k (H) were first studied in the 1980s for rings of integers in algebraic number fields ( [8,19]). Since then these invariants have been studied in a variety of settings, including numerical monoids, monoids of modules, noetherian and Krull domains (for a sample out of many we refer to [10,4,3,15,1]). …”

On elasticities of locally finitely generated monoids

“…Then all localizations R p are finitely primary and if R is not semilocal, then R has a regular element with is not a unit whence B(G P ) = {1}. For an extended list of examples, we refer to [19,Examples 5.7]. H) H such that the localization H p is finitely primary for each minimal prime ideal p ∈ X(H).…”

“…= B(G 0 , T, ι) = {S t ∈ F(G 0 )×T | σ(S) + ι(t) = 0 } ⊂ F(G 0 )×T = Fthe T -block monoid over G 0 defined by ι . For details about T -block monoids, see[19, Section 4].Let D be another monoid. A homomorphism φ : H → D is said to be• divisor homomorphism if φ(u) | φ(v) implies that u | v for all u, v ∈ H.…”