Completion of a Prüfer domain

Abstract

Let V (resp. D) be a valuation domain (resp. SFT Prüfer domain), I a proper ideal, and V̂ (resp. D̂) be the I-adic completion of V (resp. D). We show that (1) V̂ is a valuation domain, (2) Krull dimension of V ̂ = dimV/I+1 if I is not idempotent, V ̂ ≅ V/I if I is idempotent, (3) dim D ̂ = dim D/I+1 , (4) D̂ is an SFT Prüfer ring, and (5) D̂ is a catenarian ring.