Completion of a Prüfer domain (Q1300644)

Let \(V\) be a valuation domain and \(I\) be a proper ideal of \(V\). In the article are investigated properties of \(I\)-adic completion \(\widehat V\) of \(V\). Namely it is proved that (1) \(\widehat V\) is a valuation domain and its value group is isomorphic to the valuation group of \(V/\bigcap_{n \in\mathbb{N}} I^n\). (2) The Krull dimension of \(\widehat V\) is equal to \(\dim V/I+1\) if \(I\) is not idempotent. Otherwise \(V\) is isomorphic to \(V/I\). (3) If \(V\) is a Prüfer SFT domains (i.e. \(V\) is a Prüfer domain and for every ideal \(J\) in \(V\) there exists a finitely generated ideal \(I\) in \(V\) and \(n\in\mathbb{N}\) such that \(j^n\in I\) for every \(j\in J)\), then \(\widehat V\) is an SFT Prüfer ring and \(\widehat V\) is a Prüfer domain iff \(\sqrt I\) is a prime ideal.