On the structure of ⋆-power conductor domains

Abstract

Let D be an integral domain and ⋆ a star operation defined on D. We say that D is a ⋆-power conductor domain (⋆-PCD) if for each pair a, b ∈ D\(0) and for each positive integer n we have Da n ∩ Db n = ((Da ∩ Db) n ) ⋆ . We study ⋆-PCDs and characterize them as root closed domains satisfying ((a, b) n ) −1 = (((a, b) −1 ) n ) ⋆ for all nonzero a, b and all natural numbers n ≥ 1. From this it follows easily that Prüfer domains are d-PCDs (where d denotes the trivial star operation), and v-domains (e.g., Krull domains) are v-PCDs. We also consider when a ⋆-PCD is completely integrally closed, and this leads to new characterizations of Krulll domains. In particular, we show that a Noetherian domain is a Krull domain if and only if it is a w-PCD.Date: May 15, 2019. MSC 2010: 13A05, 13A15, 13E05, 13F05, 13G05.