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

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.