A localization of a power series ring over a Prüfer domain
An integral domain \(D\) is called Strong Finite Type domain (SFT-domain) if every ideal \(I\) of \(D\) is an SFT-ideal, that is there exist a finitely generated ideal \(J \subseteq I\) and a positive integer \(k\) such that \(a^k \in J\) for each \(a \in I\). A domain \(D\) is called piecewise Noetherian if it has Noetherian prime spectrum and satisfies the ascending chain condition on \(P\)-primary ideals for each prime ideal \(P\) of \(D\). These two properties are both Noetherian-like property. This paper is mainly interested in the study of the properties of various localizations of a power series ring over Prüfer domain. The author investigates the case where \(K[\![X]\!]\) and \(D[\![X]\!]\) have the same quotient field where \(D\) is either a valuation domain or an SFT- Prüfer domain. The author also investigates when \(D[\![X]\!]_{P[\![X]\!]}\) is Noetherian, where \(P\) is a nonzero SFT-prime ideal of \(D\). Finally, the author characterizes when \(D[\![X]\!]\) is piecewise Noetherian.
scientific article; zbMATH DE number 7371432