On Mockor's question

For certain classes of Prufer domains A, we study the completion (A) over c ap'(T) of A with respect to the supremum topology T = sup{T-w\w is an eleme nt of Omega}, where Omega is the family of nontrivial valuations on the quo tient field which are nonnegative on A and F-w is a topology induced by a v aluation w is an element of Omega. It is shown that the concepts "SFT Prufe r domain" and "generalized Dedekind domain" are the same. We show that if E is the ring of entire functions, then (E) over cap(,T) is a Bezout ring wh ich is not a (T) over cap-Prufer ring, and if A is an SFT Prufer domain, th en (A) over cap(,T) is a Priifer ring under a certain condition. We also sh ow that under the same conditions as above, (A) over cap(,T) is a (T) over cap-Prufer ring if and only if the number of independent valuation overring s of A is finite. In particular, if A is a Dedekind domain (resp., h-local Priifer domain), then (A) over cap(,T) is a (T) over cap Prufer ring if and only if A has only finitely many prime ideals (resp., maximal ideals). The se provide an answer to Mockor's question. (C) 1999 Academic Press.