A localization of a power series ring over a Prüfer domain

Abstract

Let D be an SFT Prüfer domain with quotient field K. We give a characterization, in terms of prime ideals of D, of when K〚X〛=D〚X〛D∖(0) holds. We next show that for a nonzero prime ideal P of D, the following statements are equivalent: (1) D〚X〛P〚X〛 is Noetherian; (2) htP=1 and k¯〚X〛=D¯〚X〛D¯∖(0), where D¯=D/P and k¯ is the quotient field of D¯; (3) D〚X〛P〚X〛 is a valuation domain. As a corollary, we give an easy proof of that D〚X〛 is piecewise Noetherian if and only if D is Noetherian. Also, we characterize nonzero prime ideals Q of D〚X1,…,Xn〛 for which D〚X1,…,Xn〛Q is a Noetherian ring, a valuation domain, or a Krull domain.