Completion of a Prufer domain

Let V (resp. D) be a valuation domain (resp. SFT Prufer domain), I a proper ideal, and (V) over cap (resp. (D) over cap) be the I-adic completion of V (resp. D). We show that (1) (V) over cap is a valuation domain, (2) Krull dimension of (V) over cap dim V/I + 1 if I is not idempotent, (V) over cap congruent to V/I if I is idempotent, (3) dim (D) over cap = dimD/I + 1, (4) (D) over cap is an SFT Prufer ring, and (5) (D) over cap is a catenarian ri ng. (C) 1999 Elsevier Science B.V. All rights reserved.