The Japanese and universally Japanese properties for valuation rings and Prüfer domains

TL;DR

The paper extends the Japanese (N-2) and universally Japanese properties to non-Noetherian contexts, proving that for valuation rings and Prüfer domains these notions coincide. It develops a valuation-theoretic criterion for N-2, a Serre-type normality framework, and a finite, dualizing-like construction over Prüfer bases to connect local data to global universal Japanese behavior. The results yield broad families of non-Noetherian universally Japanese rings, including absolute integral closures, while also providing precise counterexamples that delineate the limits of the equivalences. The work thus generalizes Nagata's Noetherian perspective to a wide class of non-Noetherian rings and clarifies when finiteness of integral closures governs universal behavior in valuation-theoretic and Prüfer settings.

Abstract

We discuss the Japanese and universally Japanese properties for valuation rings and Prüfer domains. These properties, regarding finiteness of integral closure, have been studied extensively for Noetherian rings, but very rarely, if ever, for non-Noetherian rings. Among other results, we show that for valuation rings and Prüfer domains, the Japanese and universally Japanese properties are equivalent. This result can be seen as a counterpart to Nagata's classical result for Noetherian rings. This result also tells us many non-Noetherian rings, including all absolutely integrally closed valuation rings and Prüfer domains, are universally Japanese.

Theorems & Definitions (51)

• Definition 1
• Example 2
• Example 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Remark 8
• Remark 9
• Theorem 10